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Geometry of midpoint sets for Thompson’s metric
Linear Algebra and its Applications 439 (2013) 211–227
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Geometry of midpoint sets for Thompson’s metric Yongdo Lim Department of Mathematics, Sungkyunkwan University, Suwan 440-746, Republic of Korea
ARTICLE INFO
ABSTRACT
Article history: Received 23 November 2012 Accepted 5 March 2013 Available online 6 April 2013
It is well-known that the convex cone Pm of m × m positive definite matrices is a Cartan–Hadamard Riemannian manifold with respect to the Riemannian trace metric where the geometric mean of two positive definite matrices coincides with the unique metric midpoint between them. In this paper we consider the Thompson metric on Pm inherited from the spectral norm and study some geometric structures of the Thompson midpoints. We prove that there is a unique midpoint (minimal geodesic) between A and B if and only if the spectrum of A−1 B is contained in {a, a−1 } for some a > 0, and the set of Thompson midpoints between A and B is compact and is convex in both Riemannian and Euclidean sense. It is further shown that the set of all weighted midpoints between A and B is compact and convex in Riemannian sense. © 2013 Elsevier Inc. All rights reserved.
Submitted by B. Lemmens AMS classification: 15A48 48B40 53C22 Keywords: Positive definite matrix Unitarily invariant norm Invariant metric Midpoint Minimal geodesic Convexity Thompson metric
1. Introduction The open convex cone Pm of m × m positive definite Hermitian matrices of the space Hm of Hermitian matrices has rich geometric structures inherited from symmetric gauge norms (equivalently, unitarily invariant norms). For a symmetric gauge norm on Rm , the norm on Hm defined by
||A|| := (λ1 (A), λ2 (A), . . . , λm (A)), where λj (A) are the (decreasingly ordered) eigenvalues of A ∈ Hm , induces a GL (m, C)-invariant Finsler metric on Pm and the corresponding metric distance between positive definite matrices A and B is d (A, B) = || log A−1/2 BA−1/2 || . One of common properties of these geometries is that the curve E-mail address: [email protected] 0024-3795/$ - see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2013.03.012
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Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
t → A#t B acts as a minimal geodesic from A to B for d , where A#t B = A1/2 (A−1/2 BA−1/2 )t A1/2 and is known as the t-weighted geometric mean of A and B. An immediate consequence is that the geometric mean A#B = A#1/2 B is a metric midpoint of between A and B for any d . See [3,5] for details. The Frobenius norm || · ||2 gives rise to the Riemannian structure X , Y A = Tr(A−1 XA−1 Y ), where A ∈ Pm , X , Y ∈ TA (Pm ) = Hm and then Pm becomes a Cartan–Hadamard manifold, a simply connected complete Riemannian manifold with non-positive sectional curvature [13,3,5]. In this case, there is a unique (minimal geodesic) metric midpoint between A and B, which coincides with the geometric mean A#B (t → A#t B). A subset ⊂ Pm is said to be geodesically convex if A#t B ∈ for all t ∈ [0, 1], whenever A, B ∈ . The existence of a unique minimizer (best approximant) from a closed and geodesically convex set to a given point with respect to the Riemannian distance is wellknown and the notion of convexity is useful in the study of geometric analysis and in a variety of metric-based computational algorithms for positive definite matrices; e.g., interpolation, filtering, estimation, subdivision schemes, optimization and averaging, where it has been increasingly recognized that the Euclidean distance is often not the most suitable for the set of positive definite matrices and that working with the appropriate geometry does matter in computational problems [1,9,11,17,22]. In general, the midpoint and the minimal geodesic between two points in Pm are not unique under d . For instance, there are infinitely many (minimal geodesics) midpoints between a certain pair of positive definite matrices with respect to the Finsler metric inherited from the spectral norm || · ||∞ , which coincides with Thompson’s part metric d∞ (A, B)
= || log A−1/2 BA−1/2 ||∞ = max{log λ1 (A−1/2 BA−1/2 ), log λ1 (A1/2 B−1 A1/2 )}.
In this paper we introduce the t-weighted d -midpoints between A and B; M (t ; A, B) = {X ∈ Pm : d (A, X ) = td (A, B), d (B, X ) = (1 − t )d (A, B)} and M (A, B) = t ∈[0,1] M (t ; A, B), the set of all weighted midpoints between A and B, called the metric interval between A and B. These are indeed natural extensions of the notions of Riemannian metric midpoint (geometric mean) and the Riemannian geodesic (weighted geometric mean curve) into the non-Riemannian setting of d . Mathematically, there are two main questions.
1. Under what conditions does a given pair of positive definite matrices (A, B) allow a unique (resp. multiple) midpoint or minimal geodesic from A to B? 2. If there are many midpoints (minimal geodesics) between A and B, what are the geometric structures of M (t ; A, B) and M (A, B)? Are they geodesically convex? The focus of this work is on these questions, which has not previously been investigated in any depth. We show that M (t ; A, B) and M (A, B) are compact and geodesically convex. It is shown that the t-weighted midpoints for the Thompson metric arises as an intersection of two matrix Löwner order intervals. We also obtain the following uniqueness criterion for the Thompson midpoint (minimal geodesic); there is a unique d∞ -midpoint (minimal geodesic) between A and B if and only if σ (A−1 B) ⊆ {a, a−1 } for some a > 0, where σ (X ) denotes the spectrum of X. 2. Unitarily invariant norms and Finsler metrics Let Mm be the space of m × m complex matrices with the inner product X , Y = tr(X ∗ Y ), Hm the space of n × n complex Hermitian matrices, and Pm the convex cone of positive definite Hermitian matrices. The general linear group GL (m, C) acts on Pm via congruence transformations C (X ) = CXC ∗ . For X , Y ∈ Hm , we write that X Y if Y − X is positive semidefinite, and X < Y if Y − X is positive definite. For X ∈ Hm , we denote λj (A) by j-th (decreasingly ordered) eigenvalue of X. We note that for A, B ∈ Pm , λj (A−1/2 BA−1/2 ) = λj (A−1 B) for all j = 1, 2, . . . , m from their similarity. For A, B ∈ Pm and t ∈ R, the t-weighted geometric mean of A and B is defined by A#t B
= A1/2 (A−1/2 BA−1/2 )t A1/2 .
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
213
The following properties for the weighted geometric mean are well-known [12,15]. Lemma 2.1. Let A, B, C , D (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x)
∈ Pm and let t ∈ R. Then
A#t B = A1−t Bt for AB = BA; (aA)#t (bB) = a1−t bt (A#t B) for a, b > 0; (Löwner–Heinz inequality) A#t B C#t D for A C , B D and t ∈ [0, 1]; M (A#t B)M ∗ = (MAM ∗ )#t (MBM ∗ ) for non-singular M ; A#t B = B#1−t A, (A#t B)−1 = A−1 #t B−1 ; (λA + (1 − λ)B)#t (λC + (1 − λ)D) λ(A#t C ) + (1 − λ)(B#t D) for λ, t det(A#t B) = det(A)1−t det(B)t ; ((1 − t )A−1 + tB−1 )−1 A#t B (1 − t )A + tB for t ∈ [0, 1]; (A#t B)#s (A#u B) = A#(1−s)t +su B for any s, t , u ∈ R; (Riccati Lemma) A#B is a unique positive definition solution of XA−1 X = B.
∈ [0, 1];
A norm on Rm is called a symmetric gauge function if it is invariant under permutations and sign changes of coordinates. Every symmetric gauge function induces a unitarily invariant norm on Hm
||A|| := (λ1 (A), λ2 (A), . . . , λm (A)), and conversely all unitarily invariant norms arise in this way by a theorem of von Neumann. For m 1 p p , (1 p < ∞) and ∞ (x) = max{|xi | : 1 i m} are instance, p (x) := i=1 |xi | symmetric gauge functions that induce the Schatten p-norm ⎛
||A||p = ⎝
n i=1
⎞1/p p⎠
λi (A)
and the spectral norm ||A||∞ = λ1 (A) = ||A||, respectively. Let be a symmetric gauge norm. We define a norm on the tangent space to Pm at A ∈ Pm , which is TA (Pm ) = {A} × Hm ≡ Hm , by ||X ||A = ||A−1/2 XA−1/2 || . This leads a Finsler metric on Pm . For a path γ : [0, 1] → Pm , we define its length as 1 L (γ ) = ||γ −1/2 (t )γ (t )γ −1/2 (t )|| dt (2.1) 0
and for A, B
∈ Pm , its distance
d (A, B)
= inf {L (γ ) : γ is a path from A to B}.
(2.2)
In [18], Nussbaum has studied in detail the Finsler metric from the spectral norm in general setting of normed spaces. Bhatia [3] has found some important Finsler structures on Pm . Theorem 2.2 [3,5]. We have d (A, B) = || log(A−1/2 BA−1/2 )|| and d is a complete metric distance on Pm such that for A, B ∈ Pm and M ∈ GL (m, C), (i) (ii) (iii) (iv)
d (A, B) = d (A−1 , B−1 ) = d (MAM ∗ , MBM ∗ ); d (A#B, A) = d (A#B, B) = 12 d (A, B); d (A#t B, A#s B) = |s − t |d (A, B) for all t , s ∈ [0, 1]; d (At , Bt ) td (A, B), t ∈ [0, 1].
Theorem 2.3. For A, B, C , D d (A#s B, C#t D)
∈ Pm and for s, t ∈ [0, 1],
(1 − t )d (A, C ) + |t − s|d (A, D) + sd (B, D).
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Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
In particular, d (A#t B, C#t D)
(1 − t )d (A, C ) + td (B, D), t ∈ [0, 1].
(2.3)
Proof. By Theorem 2.2, d (A#t B, A#t C ) = d (A1/2 (A−1/2 BA−1/2 )t A1/2 , A1/2 (A−1/2 CA−1/2 )t A1/2 )
= d ((A−1/2 BA−1/2 )t , (A−1/2 CA−1/2 )t ) td (A−1/2 BA−1/2 , A−1/2 CA−1/2 ) = td (B, C ) for all A, B, C
> 0 and t ∈ [0, 1]. By the triangular inequality and Lemma 2.1(v)
d (A#t B, C#t D) = d (B#1−t A, C#t D)
d (B#1−t A, B#1−t C ) + d (B#1−t C , C#t D) = d (B#1−t A, B#1−t C ) + d (C#t B, C#t D) (1 − t )d (A, C ) + td (B, D). Next, let 0
s, t 1. We first note that d (A#s B, A#s D) sd (B, D) and
d (A#t D, C#t D) and d (A#s D, A#t D)
= d (D#1−t A, D#1−t C ) (1 − t )d (A, C )
= |t − s|d (A, D) by Theorem 2.2. The previous inequalities yields
d (A#s B, C#t D) d (A#s B, A#s D) + d (A#s D, A#t D) + d (A#t D, C#t D)
sd (B, D) + |t − s|d (A, D) + (1 − t )d (A, C ).
Remark 2.4. We observe that the metric d is invariant under the congruence transformations and the matrix inversion. Furthermore, the metric topology (Pm , d ) coincides with the norm topology of Pm by its Finsler structure ((2.1) and (2.2)). Remark 2.5. The generalized exponential metric increasing property (EMI) (see [5], page 223) says that d (A, B)
|| log A − log B||
for all A, B ∈ Pm . This ensures that each closed ball for d centered at A in Pm is contained in the image of the || · || ball centered at log A in Hm under the exponential mapping. Definition 2.6. For A, B ∈ Pm , a point X in Pm is called a midpoint between A and B with respect to d if d (A, X ) = d (B, X ) = 12 d (A, B). A map f : [0, 1] → Pm is called a minimal geodesic with respect to d from A to B if d (f (s), f (t )) for all s, t
= |s − t |d (A, B), f (0) = A, f (1) = B
∈ [0, 1].
Remark 2.7. We note that if f is a minimal geodesic from A to B, then g (t ) := f (1 − t ) is a minimal geodesic from B to A and f (1/2) is a midpoint between A and B. By Theorem 2.2(iii), the curve t → A#t B is a minimal geodesic and A#B is a midpoint between A and B for any d . Then by Theorem 2.3, the metric space (Pm , d ) equipped with distinguished minimal geodesics of the form t → A#t B is non-positively curved in the sense of Busemann [7,6].
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
Lemma 2.8. If f
: [0, 1] → Pm satisfies
d (f (t ), f (s)) for all s, t
215
|t − s|d (f (0), f (1))
∈ [0, 1], then f is a minimal geodesic.
Proof. Let s, t
∈ [0, 1] with s < t. Then
d (f (0), f (1)) d (f (0), f (s)) + d (f (s), f (t )) + d (f (t ), f (1))
(s + t − s + (1 − t ))d (f (0), f (1)) = d (f (0), f (1)) which implies that d (f (s), f (t ))
= (s − t )d (f (0), f (1)).
The following provides a method of constructing infinitely many distinct minimal geodesics from two distinct minimal geodesics. Proposition 2.9. Let fi : [0, 1] → Pm , i Then for any u ∈ [0, 1], f1 #u f2 : [0, 1] →
= 1, 2, be minimal geodesics from A to B with respect to d . Pm defined by
(f1 #u f2 )(t ) = f1 (t )#u f2 (t ) is a minimal geodesic from A to B with respect to d . Furthermore if f1 (t ) (f1 #u f2 )(t ) = (f1 #v f2 )(t ) for any distinct u, v ∈ [0, 1]. Proof. Let f1 , f2 be minimal geodesic from A and B, and let u d ((f1 #u f2 )(t ), (f1 #u f2 )(s))
= f2 (t ) for some t ∈ (0, 1), then
∈ [0, 1]. Then for s, t ∈ [0, 1],
= d (f1 (t )#u f2 (t ), f1 (s)#u f2 (s)) (2.3)
(1 − u)d (f1 (t ), f1 (s)) + ud (f2 (t ), f2 (s))
= ((1 − u)|t − s| + u|t − s|)d (A, B) = |t − s|d (A, B). By Lemma 2.8, f1 #u f2 is a minimal geodesic from A to B. The remaining part of proof follows from the fact that if A = B, then A#t B = A#s B for any t = s. Indeed, A#t B = A#s B implies that (A−1/2 BA−1/2 )t = (A−1/2 BA−1/2 )s , which yields A−1/2 BA−1/2 = I. Example 2.10 (Riemannian metric). The Frobenius norm || · ||2 gives rise to the Riemannian structure
X , Y A = Tr(A−1 XA−1 Y ), where A ∈ Pm , X , Y ∈ TA (Pm ) = Hm . In this case, the curve t → A#t B is the unique (minimal) geodesic from A to B and A#B is a unique midpoint between A and B. This holds true for Schatten p-norms, 1 < p < ∞, but not for p = 1 or p = ∞ [13,3,14]. Example 2.11 (Thompson metric). For p = ∞, d∞ (A, B) norm || · ||∞ coincides with the Thompson metric d∞ (A, B) where M (B/A) that
= log(A−1/2 BA−1/2 )∞ from the spectral
= max{log M (B/A), log M (A/B)},
= inf {α > 0 : B α A} = λ1 (A−1/2 BA−1/2 ) = λ1 (A−1 B). See [20,21,8,15]. We note
d∞ (A, B) = max{log λ1 (A−1 B), log λ1 (B−1 A)}
= max{log λ1 (A−1 B), − log λm (A−1 B)}.
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Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
We recall Nussbaum’s construction of minimal geodesics for the Thompson metric. Theorem 2.12 [18]. Let A, B 1 (i) If λ− m
∈ Pm and let λk = λk (A−1 B), k = 1, 2, . . . , m.
1 λ1 , then for any α ∈ [λ− 1 , λm ],
fα (t )
=
⎧ t t t t ⎪ ⎨ λ1 −α B + λ1 α −αλ1 A,
λ1 = λm
⎪ ⎩ λt A,
λ1 = λm
λ1 −α
λ1 −α
1
is a minimal geodesic from A to B with respect to the Thompson metric. 1 −1 (ii) If λ1 λ− m , then for any α ∈ [λ1 , λm ], ⎧ t t t t ⎪ ⎨ α −λm B + αλm −λm α A, λ1 = λm α−λm α−λm gα (t ) = ⎪ ⎩ λt A, λ1 = λm 1 is a minimal geodesic from A to B with respect to the Thompson metric. Remark 2.13. By considering the extremal cases of α , the maps ⎧ t t t t ⎪ ⎨ λ1 −λm B + λ1 λm −λm λ1 A, λ1 = λm λ1 −λm λ1 −λm φ(t ; A, B) = ⎪ ⎩ λt A, λ1 = λm , 1 and
ψ(t ; A, B) =
⎧ t −t λ1 −λ1 λ11−t −λ1t −1 ⎪ ⎪ −1 B + 1 A, ⎪ ⎪ λ −λ λ1 −λ− 1 1 ⎪ ⎨ −t 1 t t −1 1−t λm −λm λ −λ B + m−1 m A, 1 ⎪ λ− λm −λm ⎪ m −λm ⎪ ⎪ ⎪ ⎩ t λ1 A,
1 λ1 λ− m , λ1 = λm 1 λ1 λ− m , λ1 = λm
λ1 = λm
are minimal geodesics from A to B.
3. New midpoint operations on Pm We have seen in Remark 2.7 that the geometric mean operation (A, B) → A#B is a midpoint operation for any d . In this section we introduce new midpoint operations with respect to the Thompson metric. Theorem 3.1. Let mi min{a, b} for all a, b
: (0, ∞) × (0, ∞) → (0, ∞), i = 1, 2, be maps satisfying
mi (a, b) max{a, b}
> 0 and i = 1, 2. Then the map M : P2m → Pm defined by
M (A, B)
⎧ ⎪ ⎪ ⎪ ⎨√
√
⎪ ⎪ ⎩√
√1
=⎪
λ1 +
1 −1
m1 (λ1
λm +
where λk
(3.4)
,λm ) −1
m2 (λ1 ,λm
)
B+ B+
1 λ1 m1 (λ− 1 , λm )A , 1 λm m2 (λ1 , λ− m )A ,
1 λ− m λ1 1 λ1 λ− m ,
= λk (A−1 B), is a d∞ -midpoint operation on Pm . In particular, the maps
A∗B=
1
√ √ B + λ1 λm A , λ1 + λm
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
AB
⎧ √ ⎪ ⎨ λ1 (A + B), 1+λ
=⎪
√ 1 ⎩ λm (A + B), 1+λm
217
1 λ− m λ1 1 λ1 λ− m ,
are d∞ -midpoint operations on Pm . Furthermore 1 (i) A ∗ B = A B if and only if either λ1 = λm or λ1 = λ− m ; − 1/2 − 1/2 (ii) A#B = A ∗ B if and only if A BA has the spectrum {λ1 , λm }; −1 (iii) A#B = A B if and only if A−1/2 BA−1/2 has the spectrum {λ1 , λ1 }, and in this case
√ √ λ1 λm A#B = A ∗ B = A B = (A + B) = (A + B); 1 + λ1 1 + λm
(iv) A ∗ B = B ∗ A and √ A B = B A; (v) (aA) ∗ (bB)= ab(A ∗ B) for any positive reals a and b;
(vi) A ∗ B
1 2
A+ √
1
λ1 λm
(vii) (MAM ∗ ) ∗ (MBM ∗ ) matrix M.
B and A B
12 (A + B);
= M (A ∗ B)M ∗ and (MAM ∗ ) (MBM ∗ ) = M (A B)M ∗ for any non-singular √
Proof. Let A, B ∈ Pm . Suppose that λ1 = λm . Then B = λ1 A and hence M (A, B) = λ1 A = A#B is −1 1 a d∞ -midpoint between A and B. Suppose that λ1 = λm . If λ− m λ1 , then by (3.4), m1 (λ1 , λm ) ∈
1 −1 [λ− 1 , λm ]. By Remark 2.13, M (A, B) is a d∞ -midpoint between A and B. Similarly for the case λm λ1 . On can see that if m1 (a, b) = max{a, b} and m2 (a, b) = min{a, b}, then M (A, B) = A ∗ B. Similarly if m1 (a, b) = min{a, b} and m2 (a, b) = max{a, b}, then M (A, B) = A B. Therefore A ∗ B and A B
are d∞ -midpoints between A and B. Next, we will check that A ∗ B and A B satisfies (i)–(vii). √ 1 λ1 λm B (i) For λ− m λ1 , A ∗ B = A B if and only if 1 −
1 −1 λ1 = λ− m or λ1 = λm . Similarly for λ1 λm .
(ii) We consider a diagonalization of A−1/2 BA−1/2 D = diag(λ1 , . . . , λm ). Since A#B = A1/2 (A−1/2 BA−1/2 )1/2 A1/2
1/2
√ √ B + λ1 λm A = A λ1 + λm
1/2
=A
D1/2
√
1
1√
λ1 + λm
B+
1
A−1/2 BA−1/2
1
UDU ∗
√ √ λ1 + λm
1/2
=
= UDU ∗ , where U is a unitary matrix and
√ √ λ1 + λm
=A
we have that A#B
√ λ1 λm A if and only if
= A1/2 (UDU ∗ )1/2 A1/2 = A1/2 UD1/2 U ∗ A1/2
and 1
= λ1 1 −
U
1
+ λ1 λm Im
+ λ1 λm Im
√ √ D + λ1 λm Im λ1 + λm
A1/2
A1/2
U ∗ A1/2 ,
√ λ1 λm A if and only if
√ =√ (3.5) D + λ1 λm Im . λ1 + λm √ √ √ √ √ Since x2 − λ1 + λm x + λ1 λm = 0 if and only if x ∈ λ1 , λm , (3.5) is equivalent to λk ∈ {λ1 , λm } for all k = 1, 2, . . . , m. That is, A−1/2 BA−1/2 has the spectrum {λ1 , λm }.
218
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227 1 (iii) Suppose that λ− m
λ1 . We note that λ1 1. Then one can see that A#B = A B if and −1
1 only if A−1/2 BA−1/2 has the spectrum {λ1 , λ1 }. Similarly for λ1 λ− m , A#B = A B if and only if − 1/2 − 1/2 − 1 A BA has the spectrum {λm , λm }. 1 (iv) Let βk = λk (B−1 A), k = 1, 2, . . . , m. Then β1 = λ1 (B−1 A) = λm (A−1 B)−1 = λ− m and
similarly βm
1 = λ− 1 . Therefore,
B∗A=
1
1
1
√ √ (A + (λm λ1 )−1/2 B) A + β1 βm B = −1/2 −1/2 β1 + βm λm + λ1
√ =√ B + λ1 λm A = A ∗ B . λ1 + λm Similarly A B = B A. (v) Let βk = λk ((aA)−1 (bB))
= ba λk (A−1 B) = ba λk . Then
1
√ (aA) ∗ (bB) = √ bB + β1 βm aA β1 + βm =
(b/a)1/2 √
1 √
λ1 +
ab
√
λm
bB + (b/a) λ1 λm aA
√ =√ B + λ1 λm A = λ1 + λm
√
ab(A ∗ B).
√
(vi) It follows from the arithmetic-geometric mean inequality 2 (vii) Straightforward.
λ1 λm λ1 + λm .
Remark 3.2. Classical means on positive reals satisfy (3.4). We note that in Theorem 3.1(ii), A−1/2 BA−1/2 has the spectrum {λ1 , λm } if and only if it has at most two distinct eigenvalues, which is always true for m = 2. Thus, 1 √ A#B = √ B + λ1 λ2 A
λ1 +
λ2
for any 2 × 2 positive definite matrices A and B. An alternate formula of the geometric mean of 2 × 2 positive definite matrices is given by A#B where α 2
=
√ αβ
det(α −1 A + β −1 B)
(α −1 A + β −1 B),
= detA and β 2 = detB. See [5].
4. Convexity of midpoints A complete metric space (M , δ) is called a Hadamard space if it satisfies the semiparallelogram law; for each a, b ∈ M, there exists an m ∈ M satisfying
δ 2 (m, x)
1 2
1
1
2
4
δ 2 (a, x) + δ 2 (b, x) − δ 2 (a, b)
(4.6)
for all x ∈ M. Such spaces are also called (global) CAT(0)-spaces or NPC (non-positively curved) spaces. In particular the m appearing in (4.6) is the unique metric midpoint between a and b. Furthermore there exists a unique minimal geodesic γ : [0, 1] → M between a and b; γ (0) = a, γ (1) = b and for all t ∈ [0, 1],
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
219
δ(γ (t ), γ (s)) = |s − t |δ(γ (0), γ (1)). We note that
δ 2 (x, γa,b (t )) (1 − t )δ 2 (x, a) + t δ 2 (x, b) − t (1 − t )δ 2 (a, b) for all x
(4.7)
∈ M and t ∈ [0, 1] and
δ 2 (a, c ) + δ 2 (b, d) δ 2 (b, c ) + δ 2 (a, d) + δ 2 (a, b) + δ 2 (c , d).
(4.8)
See [19]. The metric space (Pm , d2 ) is a standard example of a NPC space [13,14]. Lemma 4.1. For A, B, X ∈ d2 (X , B) = (1 − t )d2 (A, B).
Pm and t ∈ [0, 1], X = A#t B if and only if d2 (X , A) = td2 (A, B) and
Proof. Let X = A#t B. Then by (2.3), d2 (X , A) td2 (A, B) and d2 (X , B) triangular inequality, the equalities hold true. The converse implication follows from (4.7). We recall that for A, B > 0 and t means of A and B are defined by
(1 − t )d2 (A, B). By the
∈ [0, 1], the t-weighted arithmetic, geometric and harmonic
(1 − t )A + tB, A#t B, ((1 − t )A−1 + tB−1 )−1 , respectively. Definition 4.2. A closed subset ⊂ Pm is said to be geodesically convex (resp. geodesically complete) if A#t B ∈ for all t ∈ [0, 1] (resp. t ∈ R), whenever A, B ∈ . Lemma 4.3. For a closed subset of Pm , is geodesically convex if it is closed under the geometric mean operation. Proof. Suppose that is a closed subset of Pm invariant under the geometric mean operation. Let A, B ∈ and let D = {t ∈ [0, 1] : A#t B ∈ }. Then 0, 1, 1/2 ∈ D. If s, t ∈ D, then by Lemma 2.1(ix), (A#s B)#(A#t B) = A#(s+t )/2 B and thus (s + t )/2 ∈ D. Since D is a closed subset of [0, 1], we have D = [0, 1]. Remark 4.4. If is a non-empty geodesically convex subset of Pm , then a metric projection onto with respect to the Riemannian metric d2 always exists as it does in a Hilbert space (Theorem 6.2.6 [4]). Definition 4.5. A closed subset of Pm is called totally convex if it is invariant under the weighted arithmetic, geometric and harmonic means. For instance, the order interval [A, B] := {X > 0 : A X B} with 0 < A B is totally convex and is a compact subset of P. One can check the total convexity by using the Löwner–Heinz inequality (Lemma 2.1) and the order reverting property of the matrix inversion; 0 < A B implies that B−1 A−1 . Proposition 4.6. Every closed ball with respect to d is geodesically convex. In particular, the closed ball of radius r > 0 around A with respect to the Thompson metric is the order interval [e−r A, er A] and is totally convex. Proof. By (2.3), every closed d -ball is geodesically convex. It follows directly from the definition of the Thompson metric that d∞ (A, e−r A) = r = d∞ (A, er A), and d(A, X ) r for all e−r A X er A. Suppose that d(A, X ) r. Then M (A/X ), M (X /A)
er ,
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so X
er A and A er X, or equivalently e−r A X.
Remark 4.7. It is not easy to describe explicitly the d balls for general , even for d2 . The description of the Thompson balls via order intervals holds true for closed convex cones in a normed space. See Lemma 2.6.2 of [16] for the convexity of Thompson balls. We introduce weighted midpoints for d . Definition 4.8. For t
∈ [0, 1] and A, B ∈ Pm , we denote
M (t ; A, B) = {X > 0 : d (A, X ) = td (A, B), d (B, X ) M (A, B) = M (t ; A, B).
= (1 − t )d (A, B)},
t ∈[0,1]
We note that M (0; A, B) = {A}, M (1; A, B) = {B}. Since the map t geodesic from A to B, A#t B ∈ M (t ; A, B). By the triangular inequality, M (t ; A, B)
→ A#t B is a minimal
= {X > 0 : d (A, X ) td (A, B), d (B, X ) (1 − t )d (A, B)},
the intersection of two balls of radii td (A, B) and (1 − t )d (A, B) around A and B, respectively. Furthermore, M (1/2; A, B) coincides with the set of all d -midpoints of A and B. By Lemma 4.1, M2 (t ; A, B) = {A#t B} for all t ∈ [0, 1]. Theorem 4.9. For A, B convex. Furthermore, (1) (2) (3) (4) (5) (6)
∈ Pm and t ∈ [0, 1], M (t ; A, B) and M (A, B) are compact and geodesically
M (t ; A, A) = {A}; M (t ; A, B) ∩ M (s; A, B) = ∅ if s = t and A = B; M (t ; A, B) = M (1 − t ; B, A); M (t ; A, B)−1 = M (t ; A−1 , B−1 ); M M (t ; A, B)M ∗ = M (t ; MAM ∗ , MBM ∗ ) for all M for any s, t , u ∈ [0, 1], M (t ; A, B)#u M (s; A, B)
∈ GL (m, C);
⊂ M ((1 − u)t + us; A, B).
(4.9)
For the Thompson metric, each M∞ (t ; A, B) is totally convex and M∞ (t ; A, B)
where r
= [r −t A, r t A] ∩ [r t −1 B, r 1−t B],
(4.10)
= max{λ1 (A−1 B), λm (A−1 B)−1 = λ1 (AB−1 )}.
Proof. (1), (2), (3): Straightforward. (4), (5): It follows from the invariancy of the metric under inversion and congruence transformations. (6) Let X ∈ M (t ; A, B) and Y ∈ M (s; A, B). Then d (A, X#u Y ) = d (A#u A, X#u Y )
(2.3)
(1 − u)d (A, X ) + ud (A, Y )
[(1 − u)t + us]d (A, B) and d (B, X#u Y ) = d (B#u B, X#u Y )
(2.3)
(1 − u)d (B, X ) + ud (B, Y )
[(1 − u)(1 − t ) + u(1 − s)]d (A, B). It follows from (1 − u)(1 − t ) + u(1 − s)
= 1 − [(1 − u)t + us] that
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
221
∈ M ((1 − u)t + us; A, B).
X#u Y
Note that M (t ; A, B) is a closed subset of Pm contained in the ball of radius d (A, B) centered at A. By Remark 2.5, each d -ball is compact and hence M (t ; A, B) is compact. By (6), it is geodesically convex. Next, we will show that M (A, B) is compact and geodesically convex. By (6), it is geodesically convex. To show that M (A, B) is compact, it suffices to show that it is a closed subset of Pm from the fact that it is contained in the ball of radius d (A, B) centered at A. Let Xn be a sequence in M (A, B) converging to X ∈ Pm . For each n, pick tn ∈ [0, 1] such that Xn ∈ M (tn ; A, B). Passing to a subsequence, we may assume that tn → t for some t ∈ [0, 1]. Then d (A, X ) = lim d (A, Xn )
= lim tn d (A, B) = td (A, B),
d (B, X ) = lim d (B, Xn )
= lim (1 − tn )d (A, B) = (1 − t )d (A, B)
n→∞ n→∞
n→∞
n→∞
which shows that X ∈ M (A, B). Therefore, M (A, B) is a closed subset of Pm . Finally we deal with the Thompson metric. Let r = max{λ1 (A−1 B), λm (A−1 B)−1 = λ1 (AB−1 )}. Then d∞ (A, B) = d∞ (I , A−1/2 BA−1/2 ) = log r. By the fact that M∞ (t ; A, B) is the intersection of the balls of radii t log r and (1 − t ) log r around A and B and by Proposition 4.6, (4.10) holds. This implies in particular that M∞ (t ; A, B) is totally convex. Remark 4.10. From M∞ (t ; A, B)
⊂ [r −t A, r t A] ∩ [r t −1 B, r 1−t B] ⊂ [(1/r )A, rA] ∩ [(1/r )B, rB]
for all t ∈ [0, 1], we have M∞ (A, B) ⊂ [(1/r )A, rA] λ1 (AB−1 )}. In general, M∞ (A, B) is not totally convex. Corollary 4.11. Let A, B matrix.
∩ [(1/r )B, rB] where r = max{λ1 (A−1 B),
∈ Pm . Let A−1/2 BA−1/2 = UDU ∗ , where U is a unitary matrix and D is a diagonal
(1) M (t ; A, B) = A1/2 U ∗ M (t ; Im , D)UA1/2 and M (A, B) = A1/2 U ∗ M (Im , D)UA1/2 . (2) Let t0 ∈ (0, 1). Then ⎧ t ⎪ ⎨ A1/2 M (t0 ; Im , A−1/2 BA−1/2 ) t0 A1/2 , t t0 M (t ; A, B) ⊂ 1−t ⎪ ⎩ B1/2 M (t0 ; B−1/2 AB−1/2 , Im ) 1−t0 B1/2 , t t0 .
(4.11)
In particular, M (t ; Im , D)
⊂
⎧ 1 ⎨M ;I
⎩ D#
2
m, D
2(1−t ) M
2t
,
1 ;I ,D 2 m
,
t
t
1 2 1 . 2
(4.12)
(3) Let D be a diagonal matrix with positive entries. If M (t0 ; I , D) consists of diagonal matrices for some t0 ∈ (0, 1), then M (I , D) consists of diagonal matrices. (4) If M (t0 ; A, B) = {A#t0 B} for some t0 ∈ (0, 1), then the curve t → A#t B is a unique minimal geodesic with respect to d from A to B. Proof. (1) It follows from Theorem 4.9. (2) Let t0 ∈ (0, 1) and let t t0 . Let X B# 1−t0 X 1−t
Since B#u X
∈ M (t ; A, B). Then by Theorem 4.9(6),
∈ M (1; A, B)# 1−t0 M (t ; A, B) ⊂ M (t0 ; A, B).
=B
1−t
1/2
(B−1/2 XB−1/2 )u B1/2 , we have
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Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
X
1−t
1−t
∈ B1/2 [B−1/2 M (t0 ; A, B)B−1/2 ] 1−t0 B1/2 = B1/2 M (t0 ; B−1/2 AB−1/2 , Im ) 1−t0 B1/2 . 1−t
Therefore M (t ; A, B) ⊂ B1/2 M (t0 ; B−1/2 AB−1/2 , Im ) 1−t0 B1/2 . Suppose that t t0 . Then 1 − t 1 − t0 and by the preceding step, M (t ; A, B) = M (1 − t ; B, A) t
⊂ A1/2 M (1 − t0 ; A−1/2 BA−1/2 , Im ) t0 A1/2 t
= A1/2 M (t0 ; Im , A−1/2 BA−1/2 ) t0 A1/2 . Next, we will show (4.12) for t M (t ; Im , D) ⊂ D
1/2
12 . By (4.11),
M (1/2; D
−1
, Im )2(1−t ) D1/2
= D1/2 (Im #2(1−t ) M (1/2; D−1 , Im ))D1/2 = D#2(1−t ) M (1/2; Im , D)). (3) It follows from (4.11). (4) Suppose that M (t0 ; A, B) M (t0 ; A, B) = A
1/2
= {A#t0 B} for some t0 ∈ (0, 1). By (1), −1/2
M (t0 ; Im , A
BA−1/2 )A1/2
= {A#t0 B}
= A1/2 {(A−1/2 BA−1/2 )t0 }A1/2 which shows that M (t0 ; I , A−1/2 BA−1/2 ) M (t0 ; A, B) = M (1 − t0 ; B, A)
= {(A−1/2 BA−1/2 )t0 }. Similarly
= B1/2 M (1 − t0 ; Im , B−1/2 AB−1/2 )B1/2
= {A#t0 B} = {B#1−t0 A} = B1/2 {(B−1/2 AB−1/2 )1−t0 }B1/2 yields that M (t0 ; B
−1/2
AB−1/2 , Im )
= M (1 − t0 ; Im , B−1/2 AB−1/2 ) = {(B−1/2 AB−1/2 )1−t0 }.
By (4.11), the proof is completed. 5. Uniqueness of d∞ -minimal geodesic In this section we find a uniqueness criterion for minimal geodesics with respect to the Thompson metric. We denote σ (A) by the spectrum of A. We note that for Hermitian matrices A and B, A is similar to B if and only if A is unitarily similar to B if and only if λi (A) = λi (B) for all 1 i m. We denote A ∼ B if and only if A is similar to B. Proposition 5.1. Let A, B ∈ Pm such that σ (A) ⊆ {a, b} for some a (unitarily) similar to A1/2 and A−1/2 respectively, then B = A1/2 .
√
b > 0. If B and A−1/2 BA−1/2 are
Proof. Suppose that a = b. Then A = aIm and hence B = aIm = A1/2 . Suppose that σ (A) = {a, b} where a > b > 0. Pick a unitary matrix U such that A = UDU ∗ where D is a diagonal matrix. Setting X = U ∗ BU, X is unitarily similar to D1/2 and D−1/2 XD−1/2 is unitarily similar to D−1/2 . Indeed, X ∼ B ∼ A1/2 = UD1/2 U ∗ ∼ D1/2 and D−1/2 XD−1/2 = (U ∗ A−1/2 U )(U ∗ BU )(U ∗ A−1/2 U ) ∼ A−1/2 BA−1/2 ∼ A−1/2 ∼ D−1/2 . If X = D1/2 , then B = UXU ∗ = UD1/2 U ∗ = A1/2 . So, it is enough to show that X = D1/2 . By assumption, D has the spectrum {a, b}.
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
223
Pick a permutation matrix P such that
= PDP = diag(a, . . . , a, b, . . . , b),
D1
m−k
k
1/2
= PXP, we have X1 ∼ PD1/2 P = D1 and D1−1 X1 = 1/2 1/2 . If X1 = D1 , then X = PX1 P = PD1 P = D1/2 . Therefore,
where k denotes the number of a’s in D. Setting X1 PD−1 PPXP
PD−1 XP
=
∼D
−1/2
1/2
∼ D1
we may assume that D
= diag(a, . . . , a, b, . . . , b).
k
m−k
Write X = [xij ]m×m ∈ Pm . Let λk = λk (X ) and βk = λk (D D−1/2 XD−1/2 are similar to D1/2 and D−1/2 respectively,
λi =
√
(1 i k),
a 1
βi = √
λi =
= λk (D−1/2 XD−1/2 ). Since X and
(k < i m),
b
1
βi = √
(1 i m − k),
b
√
−1 X )
(m − k < i m).
a
By Hadamard’s inequality (Theorem 7.8.1 [10]), √ k √ m−k a b
=
m i=1
λi = det(X )
m i=1
xii .
(5.13)
By the spectral theorem for normal matrices (Theorem 2.5.4 of [10]), ka + (m − k)b =
m i=1
+2
xii2
1i
m
i=1
λ2i =
m
i,j=1
|xij |2 . That is,
|xij |2 .
(5.14)
We note that the right-hand side is equal to m i=1
xii2
+
|xij |2 +
(i,j)∈I11
|xij |2 +
(i,j)∈I12
|xij |2 +
(i,j)∈I21
|xij |2 ,
(i,j)∈I22
where I11
= {(i, j) : 1 i, j k, i = j}, I12 = {(i, j) : 1 i k, k < j m},
I21
= {(i, j) : k < i m, 1 j k, }, I22 = {(i, j) : k < i, j m, i = j}.
By the same argument for D−1 X, we have m−k b
+
k a
=
m i=1
+
βi2 =
1
k 1
a2
b2 (i,j)∈I 21
i=1
xii2
+
|yij |2 +
1 b2
1
m i=k+1
b2 (i,j)∈I 22
xii2
+
1 a2
|yij |2 +
(i,j)∈I11
1 a2
|yij |2
(i,j)∈I12
|yij |2 .
Combining this together with (5.14) yields m−k b
(a − b ) = (a − b ) 2
2
2
2
m i=k+1
⎛ xii2
+ (a − b ) ⎝ 2
2
(i,j)∈I21
|xij | + 2
(i,j)∈I22
⎞ 2⎠
|xij |
.
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Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
In particular, implies that m
m
(m − k)b. This together with the arithmetic-geometric mean inequality
2 i=k+1 xii
m xii
i=k+1
m−k 2
2 i=k+1 xii
m−k
√ m−k b .
(5.15)
By (5.13) and (5.15), √ k √ m−k a b
m
xii
i=1
=
k
xii
i=1
·
m
xii
i=k+1
k √ m−k b · xii . i=1
That is, k
xii
i=1
√ k a .
(5.16)
Now, we compute the trace of X and D−1 X; k
√
√
a + (m − k)
b
= Tr(X ) =
m
xii
(5.17)
i=1
and m−k
√
k
+ √ = Tr(D−1 X ) = a
b
k 1
a i=1
xii
+
m 1
b i=k+1
xii .
Combining these two equations yields m
xii
i=k+1
√ = (m − k) b
(5.18)
and hence from (5.17), k i=1
xii
√ = k a.
(5.19)
By (5.16) and (5.19), √ k a
k
xii
i=1
=
i=1 xii
=
k
=
√ k a
√
√
which implies that xii = |xij |2 = 0. 1i
That is, X
k
a for all i = 1, . . . , k. This with (5.13) and (5.18) yields m m−k m √ m−k i=k+1 xii xii = b m−k i=k+1
which implies that xii √ m−k b
k
= D1/2 .
b for k
< i m. By (5.14) we conclude that
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
225
The following is one of the main results of this paper. Theorem 5.2. Let A, B ∈ Pm . Then there is a unique d∞ -minimal geodesic from A to B if and only if there is a unique midpoint between A and B if and only if σ (A−1 B) ⊆ {a, a−1 } for some a > 0. Proof. Let A, B
∈ Pm and let λk = λk (A−1 B), k = 1, 2, . . . , m. Then it suffices to show the following.
(i) If λ1 = λm , then there is a unique d∞ -minimal geodesic (midpoint) from A to B. (ii) If A−1 B has at least three distinct eigenvalues, then there are infinitely many d∞ -minimal geodesics (midpoints) from A to B. 1 (iii) If A−1 B has two distinct eigenvalues with λ1 = λ− m , then there are infinitely many d∞ -minimal geodesics (midpoints) from A to B. 1 (iv) If A−1 B has two distinct eigenvalues with λ1 = λ− m , then there is a unique d∞ -minimal geodesic (midpoint) from A to B.
= λ1 = λm . Then B = aA. By Corollary 4.11, it suffices to show that √ M∞ (1/2; Im , aIm ) = aIm .
(i) Let a
We further assume that a > 1 because M∞ (1/2; Im , aIm ) = M∞ (1/2; Im , a−1 Im )−1 . Then d∞ (Im , aIm ) = log a. Let X ∈ M∞ (1/2; Im , aIm ) and let λk = λk (X ). 1 Suppose that λ1 λ− m . Then 1 2 = d∞ (Im , aIm ) = 2d∞ (Im , X ) = 2 max{log λ1 , λ− m } = log λ1 √ implies that λ1 = a. Since λ1 (a−1 X ) = a−1 λ1 a−1 λm = λm (a−1 X ), we have that
log a
log
√
1
d∞ (Im , aIm ) = d∞ (aIm , X ) = d∞ (Im , a−1 X ) ! " a a = log = max log a−1 λ1 , log
a=
2
λm λm √ which yields λm = a. That is, X = aIm . √ 1 1 a = d∞ (Im , X ) = log λ− Suppose that λ1 < λ− m . Then log m implies that λm = √
√1 . But this is a
impossible from the fact that log
√
a
= d∞ (aI , X ) log λm (a−1 X )−1 = log
a
λm
√ = log a a.
(ii) By Remark 2.13 and Theorem 3.1(iii), the map
φ(t ; A, B) =
λt1 − λtm λ1 λtm − λm λt1 B+ A λ1 − λm λ1 − λm
is a minimal geodesic for A to B with φ(1/2; A, B) = A#B. By Proposition 2.9, there are infinitely many minimal geodesics and midpoints from A to B. −1 1 (iii) By Theorem 2.12, if λ− m < λ1 , then for any α ∈ (λ1 , λm ), fα (t ) =
λt1 − α t λ1 α t − αλt1 B+ A λ1 − α λ1 − α
is a minimal geodesics with respect to the Thompson metric from A to B. Also, if λ1 1 any α ∈ (λ1 , λ− m ), gα (t ) =
α t − λtm αλtm − λm α t B+ A α − λm α − λm
1 < λ− m , then for
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Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
is a minimal geodesics with respect to the Thompson metric from A to B. Then one can see that A#B = fα (1/2) and A#B = gα (1/2). Therefore, there are infinitely many minimal geodesics and midpoints from A to B. (iv) We will show that {A#B} = M(1/2; A, B). By Corollary 4.11, we may assume that A = Im and B
= D = diag(a, . . . , a, a−1 , . . . , a−1 ), a > 1.
m−k
k
Let X ∈ M(1/2; Im , D). We claim that X = D1/2 . By Proposition 5.1, it is enough to show that X and D−1/2 XD−1/2 are similar to D1/2 and D−1/2 , respectively. Let λk = λk (X ) and βk = λk (D−1 X ) = λk (D−1/2 XD−1/2 ). From d∞ (I , X ) = d∞ (X , D)(= d∞ (Im , D−1/2 XD−1/2 )) = 12 log a, we have 1 √ λi , βi ∈ √ , a a for all 1 i m. Applying the inequalities λi (A)λm (B) A, B ∈ Pm (Theorem 7.10 [23]), we have $ $ # #
λm a
λ1
βm min
#
a
, aλm max
max aλm ,
$
λi
λ1 a
λi (AB) = λi (BA) λi (A)λ1 (B) for any
, aλm β1 aλ1 ,
(5.20)
βi aλi (1 i m − k),
a
#
λi
βi min aλi ,
a
λ1
(5.21)
$
a
(i > m − k).
(5.22)
By Lidskii’s theorem (Corollary III.4.6 [2]), m −k i=1 m i=1
λi (D−1 )
m i=k+1
m −k i=1
βi ,
βi = det(D−1 X ) = am−2k
We note that βi
=
from (5.21) and λm am−k
m i=k+1
or
λi
%m
i=k+1 λi
√1 for all i
λi
√1
a √1 . a
(5.23)
m i=1
λi .
(5.24)
> m − k by (5.22) and λ1
√
a. Also, βi
=
√
a for all 1
By (5.23),
√ m−k a ,
m−k
a
. Since λi
√1 for all i a
= 1, 2 . . . , m, we have
1
λi = √ , i > m − k . a
However, by (5.24) √ m−2k a
=
√ m−k √ −k a a
= am−2k
=
m i=1
k √ k−m a λi i=1
βi = am−2k
m i=1
λi = am−2k
k i=1
λi
m i=k+1
λi
i m−k
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
implies that
λi =
%k
√
i=1
a, i
λi =
√ k a . Since λi
227
√
a for all i, we have
= 1, 2, . . . , k.
This shows that λi (X )
= λi (D1/2 ) and λi (D−1 X ) = λi (D−1/2 ) for all i = 1, . . . , m.
Acknowledgement The author wishes to express his gratitude to Professor Bas Lemmens for helpful insights about convexity of Thompson balls. This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (No. 2012-005191). References [1] F. Barbaresco, Innovative tools for radar signal processing based on Cartan’s geometry of symmetric positive definite matrices and information geometry, in: Proceedings of the IEEE International Radar Conference, Rome, Italy, 2008. [2] R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics,, Springer, 1996, pp. . [3] R. Bhatia, On the exponential metric increasing property, Linear Algebra Appl. 375 (2003) 211–220. [4] R. Bhatia, J. Holbrook, Riemannian geometry and matrix geometric means, Linear Algebra Appl. 413 (2006) 594–618. [5] R. Bhatia, Positive Definite Matrices, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2007. [6] M. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, Berlin, 1999. [7] H. Busemann, B.B. Phadke, Spaces with distinguished geodesics, Monographs and Textbooks in Pure and Applied Mathematics, vol. 108, Marcel Dekker Inc., New York, 1987. [8] G. Corach, H. Porta, L. Recht, Convexity of the geodesic distance on spaces of positive operators, Illinois J. Math. 38 (1994) 87–94. [9] P.D. Fletcher, S. Joshi, Riemannian geometry for the statistical analysis of difusion tensor data, Signal Process. 87 (2007) 25–262. [10] R. Horn, C. Johnson, Matrix Analysis, Cambridge University Press, 1985. [11] U. Itai, N. Sharon, Subdivision schemes for positive definite matrices, Found. Comput. Math. http://dx.doi.org/10.1007/ s10208-012-9131-y. [12] F. Kubo, T. Ando, Means of positive linear operators, Math. Ann. 246 (1980) 205–224. [13] S. Lang, Fundamentals of Differential Geometry, Graduate Texts in Mathematics, Springer-Verlag, 1999. [14] J.D. Lawson, Y. Lim, The geometric mean, matrices, metrics, and more, Am. Math. Mon. 108 (2001) 797–812. [15] J. Lawson, Y. Lim, Metric convexity of symmetric cones, Osaka J. Math. 44 (2007) 795–816. [16] B. Lemmens, R. Nussbaum, Nonlinear Perron–Frobenius Theorey, Cambridge Tracts in Mathematics, vol. 189, Cambridge University Press, 2012. [17] X. Pennec, P. Fillard, N. Ayache, A Riemannian framework for tensor computing, Int. J. Comput. Vision 66 (2006) 41–66. [18] R.D. Nussbaum, Finsler structures for the part metric and Hilbert’s projective metric and applications to ordinary differential equations, Differ. Integral Equ. 7 (1994) 1649–1707. [19] K.-T. Sturm, Probability measures on metric spaces of nonpositive curvature, in: P. Auscher, et al (Ed.), Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Contemporary Mathematics, vol. 338, American Mathematical Society (AMS), Providence, 2003. [20] A.C. Thompson, On certain contraction mappings in a partially ordered vector space, Proc. Am. Math. Soc. 14 (1963) 438–443. [21] M. Vesentini, Invariant metrics on convex cones, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3 (1976) 671–696. [22] M. Zerai, M. Moakher, The Riemannian geometry of the space of positive-definite matrices and its application to the regularization of positive-definite matrix-valued date, J. Math. Imaging Vision 40 (2011) 171–187. [23] F. Zhang, Matrix Theory: Basic Results and Techniques, Universitext, Springer-Verlag, New York, 1999.
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Geometry of midpoint sets for Thompson’s metric Yongdo Lim Department of Mathematics, Sungkyunkwan University, Suwan 440-746, Republic of Korea
ARTICLE INFO
ABSTRACT
Article history: Received 23 November 2012 Accepted 5 March 2013 Available online 6 April 2013
It is well-known that the convex cone Pm of m × m positive definite matrices is a Cartan–Hadamard Riemannian manifold with respect to the Riemannian trace metric where the geometric mean of two positive definite matrices coincides with the unique metric midpoint between them. In this paper we consider the Thompson metric on Pm inherited from the spectral norm and study some geometric structures of the Thompson midpoints. We prove that there is a unique midpoint (minimal geodesic) between A and B if and only if the spectrum of A−1 B is contained in {a, a−1 } for some a > 0, and the set of Thompson midpoints between A and B is compact and is convex in both Riemannian and Euclidean sense. It is further shown that the set of all weighted midpoints between A and B is compact and convex in Riemannian sense. © 2013 Elsevier Inc. All rights reserved.
Submitted by B. Lemmens AMS classification: 15A48 48B40 53C22 Keywords: Positive definite matrix Unitarily invariant norm Invariant metric Midpoint Minimal geodesic Convexity Thompson metric
1. Introduction The open convex cone Pm of m × m positive definite Hermitian matrices of the space Hm of Hermitian matrices has rich geometric structures inherited from symmetric gauge norms (equivalently, unitarily invariant norms). For a symmetric gauge norm on Rm , the norm on Hm defined by
||A|| := (λ1 (A), λ2 (A), . . . , λm (A)), where λj (A) are the (decreasingly ordered) eigenvalues of A ∈ Hm , induces a GL (m, C)-invariant Finsler metric on Pm and the corresponding metric distance between positive definite matrices A and B is d (A, B) = || log A−1/2 BA−1/2 || . One of common properties of these geometries is that the curve E-mail address: [email protected] 0024-3795/$ - see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2013.03.012
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Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
t → A#t B acts as a minimal geodesic from A to B for d , where A#t B = A1/2 (A−1/2 BA−1/2 )t A1/2 and is known as the t-weighted geometric mean of A and B. An immediate consequence is that the geometric mean A#B = A#1/2 B is a metric midpoint of between A and B for any d . See [3,5] for details. The Frobenius norm || · ||2 gives rise to the Riemannian structure X , Y A = Tr(A−1 XA−1 Y ), where A ∈ Pm , X , Y ∈ TA (Pm ) = Hm and then Pm becomes a Cartan–Hadamard manifold, a simply connected complete Riemannian manifold with non-positive sectional curvature [13,3,5]. In this case, there is a unique (minimal geodesic) metric midpoint between A and B, which coincides with the geometric mean A#B (t → A#t B). A subset ⊂ Pm is said to be geodesically convex if A#t B ∈ for all t ∈ [0, 1], whenever A, B ∈ . The existence of a unique minimizer (best approximant) from a closed and geodesically convex set to a given point with respect to the Riemannian distance is wellknown and the notion of convexity is useful in the study of geometric analysis and in a variety of metric-based computational algorithms for positive definite matrices; e.g., interpolation, filtering, estimation, subdivision schemes, optimization and averaging, where it has been increasingly recognized that the Euclidean distance is often not the most suitable for the set of positive definite matrices and that working with the appropriate geometry does matter in computational problems [1,9,11,17,22]. In general, the midpoint and the minimal geodesic between two points in Pm are not unique under d . For instance, there are infinitely many (minimal geodesics) midpoints between a certain pair of positive definite matrices with respect to the Finsler metric inherited from the spectral norm || · ||∞ , which coincides with Thompson’s part metric d∞ (A, B)
= || log A−1/2 BA−1/2 ||∞ = max{log λ1 (A−1/2 BA−1/2 ), log λ1 (A1/2 B−1 A1/2 )}.
In this paper we introduce the t-weighted d -midpoints between A and B; M (t ; A, B) = {X ∈ Pm : d (A, X ) = td (A, B), d (B, X ) = (1 − t )d (A, B)} and M (A, B) = t ∈[0,1] M (t ; A, B), the set of all weighted midpoints between A and B, called the metric interval between A and B. These are indeed natural extensions of the notions of Riemannian metric midpoint (geometric mean) and the Riemannian geodesic (weighted geometric mean curve) into the non-Riemannian setting of d . Mathematically, there are two main questions.
1. Under what conditions does a given pair of positive definite matrices (A, B) allow a unique (resp. multiple) midpoint or minimal geodesic from A to B? 2. If there are many midpoints (minimal geodesics) between A and B, what are the geometric structures of M (t ; A, B) and M (A, B)? Are they geodesically convex? The focus of this work is on these questions, which has not previously been investigated in any depth. We show that M (t ; A, B) and M (A, B) are compact and geodesically convex. It is shown that the t-weighted midpoints for the Thompson metric arises as an intersection of two matrix Löwner order intervals. We also obtain the following uniqueness criterion for the Thompson midpoint (minimal geodesic); there is a unique d∞ -midpoint (minimal geodesic) between A and B if and only if σ (A−1 B) ⊆ {a, a−1 } for some a > 0, where σ (X ) denotes the spectrum of X. 2. Unitarily invariant norms and Finsler metrics Let Mm be the space of m × m complex matrices with the inner product X , Y = tr(X ∗ Y ), Hm the space of n × n complex Hermitian matrices, and Pm the convex cone of positive definite Hermitian matrices. The general linear group GL (m, C) acts on Pm via congruence transformations C (X ) = CXC ∗ . For X , Y ∈ Hm , we write that X Y if Y − X is positive semidefinite, and X < Y if Y − X is positive definite. For X ∈ Hm , we denote λj (A) by j-th (decreasingly ordered) eigenvalue of X. We note that for A, B ∈ Pm , λj (A−1/2 BA−1/2 ) = λj (A−1 B) for all j = 1, 2, . . . , m from their similarity. For A, B ∈ Pm and t ∈ R, the t-weighted geometric mean of A and B is defined by A#t B
= A1/2 (A−1/2 BA−1/2 )t A1/2 .
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
213
The following properties for the weighted geometric mean are well-known [12,15]. Lemma 2.1. Let A, B, C , D (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x)
∈ Pm and let t ∈ R. Then
A#t B = A1−t Bt for AB = BA; (aA)#t (bB) = a1−t bt (A#t B) for a, b > 0; (Löwner–Heinz inequality) A#t B C#t D for A C , B D and t ∈ [0, 1]; M (A#t B)M ∗ = (MAM ∗ )#t (MBM ∗ ) for non-singular M ; A#t B = B#1−t A, (A#t B)−1 = A−1 #t B−1 ; (λA + (1 − λ)B)#t (λC + (1 − λ)D) λ(A#t C ) + (1 − λ)(B#t D) for λ, t det(A#t B) = det(A)1−t det(B)t ; ((1 − t )A−1 + tB−1 )−1 A#t B (1 − t )A + tB for t ∈ [0, 1]; (A#t B)#s (A#u B) = A#(1−s)t +su B for any s, t , u ∈ R; (Riccati Lemma) A#B is a unique positive definition solution of XA−1 X = B.
∈ [0, 1];
A norm on Rm is called a symmetric gauge function if it is invariant under permutations and sign changes of coordinates. Every symmetric gauge function induces a unitarily invariant norm on Hm
||A|| := (λ1 (A), λ2 (A), . . . , λm (A)), and conversely all unitarily invariant norms arise in this way by a theorem of von Neumann. For m 1 p p , (1 p < ∞) and ∞ (x) = max{|xi | : 1 i m} are instance, p (x) := i=1 |xi | symmetric gauge functions that induce the Schatten p-norm ⎛
||A||p = ⎝
n i=1
⎞1/p p⎠
λi (A)
and the spectral norm ||A||∞ = λ1 (A) = ||A||, respectively. Let be a symmetric gauge norm. We define a norm on the tangent space to Pm at A ∈ Pm , which is TA (Pm ) = {A} × Hm ≡ Hm , by ||X ||A = ||A−1/2 XA−1/2 || . This leads a Finsler metric on Pm . For a path γ : [0, 1] → Pm , we define its length as 1 L (γ ) = ||γ −1/2 (t )γ (t )γ −1/2 (t )|| dt (2.1) 0
and for A, B
∈ Pm , its distance
d (A, B)
= inf {L (γ ) : γ is a path from A to B}.
(2.2)
In [18], Nussbaum has studied in detail the Finsler metric from the spectral norm in general setting of normed spaces. Bhatia [3] has found some important Finsler structures on Pm . Theorem 2.2 [3,5]. We have d (A, B) = || log(A−1/2 BA−1/2 )|| and d is a complete metric distance on Pm such that for A, B ∈ Pm and M ∈ GL (m, C), (i) (ii) (iii) (iv)
d (A, B) = d (A−1 , B−1 ) = d (MAM ∗ , MBM ∗ ); d (A#B, A) = d (A#B, B) = 12 d (A, B); d (A#t B, A#s B) = |s − t |d (A, B) for all t , s ∈ [0, 1]; d (At , Bt ) td (A, B), t ∈ [0, 1].
Theorem 2.3. For A, B, C , D d (A#s B, C#t D)
∈ Pm and for s, t ∈ [0, 1],
(1 − t )d (A, C ) + |t − s|d (A, D) + sd (B, D).
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Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
In particular, d (A#t B, C#t D)
(1 − t )d (A, C ) + td (B, D), t ∈ [0, 1].
(2.3)
Proof. By Theorem 2.2, d (A#t B, A#t C ) = d (A1/2 (A−1/2 BA−1/2 )t A1/2 , A1/2 (A−1/2 CA−1/2 )t A1/2 )
= d ((A−1/2 BA−1/2 )t , (A−1/2 CA−1/2 )t ) td (A−1/2 BA−1/2 , A−1/2 CA−1/2 ) = td (B, C ) for all A, B, C
> 0 and t ∈ [0, 1]. By the triangular inequality and Lemma 2.1(v)
d (A#t B, C#t D) = d (B#1−t A, C#t D)
d (B#1−t A, B#1−t C ) + d (B#1−t C , C#t D) = d (B#1−t A, B#1−t C ) + d (C#t B, C#t D) (1 − t )d (A, C ) + td (B, D). Next, let 0
s, t 1. We first note that d (A#s B, A#s D) sd (B, D) and
d (A#t D, C#t D) and d (A#s D, A#t D)
= d (D#1−t A, D#1−t C ) (1 − t )d (A, C )
= |t − s|d (A, D) by Theorem 2.2. The previous inequalities yields
d (A#s B, C#t D) d (A#s B, A#s D) + d (A#s D, A#t D) + d (A#t D, C#t D)
sd (B, D) + |t − s|d (A, D) + (1 − t )d (A, C ).
Remark 2.4. We observe that the metric d is invariant under the congruence transformations and the matrix inversion. Furthermore, the metric topology (Pm , d ) coincides with the norm topology of Pm by its Finsler structure ((2.1) and (2.2)). Remark 2.5. The generalized exponential metric increasing property (EMI) (see [5], page 223) says that d (A, B)
|| log A − log B||
for all A, B ∈ Pm . This ensures that each closed ball for d centered at A in Pm is contained in the image of the || · || ball centered at log A in Hm under the exponential mapping. Definition 2.6. For A, B ∈ Pm , a point X in Pm is called a midpoint between A and B with respect to d if d (A, X ) = d (B, X ) = 12 d (A, B). A map f : [0, 1] → Pm is called a minimal geodesic with respect to d from A to B if d (f (s), f (t )) for all s, t
= |s − t |d (A, B), f (0) = A, f (1) = B
∈ [0, 1].
Remark 2.7. We note that if f is a minimal geodesic from A to B, then g (t ) := f (1 − t ) is a minimal geodesic from B to A and f (1/2) is a midpoint between A and B. By Theorem 2.2(iii), the curve t → A#t B is a minimal geodesic and A#B is a midpoint between A and B for any d . Then by Theorem 2.3, the metric space (Pm , d ) equipped with distinguished minimal geodesics of the form t → A#t B is non-positively curved in the sense of Busemann [7,6].
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
Lemma 2.8. If f
: [0, 1] → Pm satisfies
d (f (t ), f (s)) for all s, t
215
|t − s|d (f (0), f (1))
∈ [0, 1], then f is a minimal geodesic.
Proof. Let s, t
∈ [0, 1] with s < t. Then
d (f (0), f (1)) d (f (0), f (s)) + d (f (s), f (t )) + d (f (t ), f (1))
(s + t − s + (1 − t ))d (f (0), f (1)) = d (f (0), f (1)) which implies that d (f (s), f (t ))
= (s − t )d (f (0), f (1)).
The following provides a method of constructing infinitely many distinct minimal geodesics from two distinct minimal geodesics. Proposition 2.9. Let fi : [0, 1] → Pm , i Then for any u ∈ [0, 1], f1 #u f2 : [0, 1] →
= 1, 2, be minimal geodesics from A to B with respect to d . Pm defined by
(f1 #u f2 )(t ) = f1 (t )#u f2 (t ) is a minimal geodesic from A to B with respect to d . Furthermore if f1 (t ) (f1 #u f2 )(t ) = (f1 #v f2 )(t ) for any distinct u, v ∈ [0, 1]. Proof. Let f1 , f2 be minimal geodesic from A and B, and let u d ((f1 #u f2 )(t ), (f1 #u f2 )(s))
= f2 (t ) for some t ∈ (0, 1), then
∈ [0, 1]. Then for s, t ∈ [0, 1],
= d (f1 (t )#u f2 (t ), f1 (s)#u f2 (s)) (2.3)
(1 − u)d (f1 (t ), f1 (s)) + ud (f2 (t ), f2 (s))
= ((1 − u)|t − s| + u|t − s|)d (A, B) = |t − s|d (A, B). By Lemma 2.8, f1 #u f2 is a minimal geodesic from A to B. The remaining part of proof follows from the fact that if A = B, then A#t B = A#s B for any t = s. Indeed, A#t B = A#s B implies that (A−1/2 BA−1/2 )t = (A−1/2 BA−1/2 )s , which yields A−1/2 BA−1/2 = I. Example 2.10 (Riemannian metric). The Frobenius norm || · ||2 gives rise to the Riemannian structure
X , Y A = Tr(A−1 XA−1 Y ), where A ∈ Pm , X , Y ∈ TA (Pm ) = Hm . In this case, the curve t → A#t B is the unique (minimal) geodesic from A to B and A#B is a unique midpoint between A and B. This holds true for Schatten p-norms, 1 < p < ∞, but not for p = 1 or p = ∞ [13,3,14]. Example 2.11 (Thompson metric). For p = ∞, d∞ (A, B) norm || · ||∞ coincides with the Thompson metric d∞ (A, B) where M (B/A) that
= log(A−1/2 BA−1/2 )∞ from the spectral
= max{log M (B/A), log M (A/B)},
= inf {α > 0 : B α A} = λ1 (A−1/2 BA−1/2 ) = λ1 (A−1 B). See [20,21,8,15]. We note
d∞ (A, B) = max{log λ1 (A−1 B), log λ1 (B−1 A)}
= max{log λ1 (A−1 B), − log λm (A−1 B)}.
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Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
We recall Nussbaum’s construction of minimal geodesics for the Thompson metric. Theorem 2.12 [18]. Let A, B 1 (i) If λ− m
∈ Pm and let λk = λk (A−1 B), k = 1, 2, . . . , m.
1 λ1 , then for any α ∈ [λ− 1 , λm ],
fα (t )
=
⎧ t t t t ⎪ ⎨ λ1 −α B + λ1 α −αλ1 A,
λ1 = λm
⎪ ⎩ λt A,
λ1 = λm
λ1 −α
λ1 −α
1
is a minimal geodesic from A to B with respect to the Thompson metric. 1 −1 (ii) If λ1 λ− m , then for any α ∈ [λ1 , λm ], ⎧ t t t t ⎪ ⎨ α −λm B + αλm −λm α A, λ1 = λm α−λm α−λm gα (t ) = ⎪ ⎩ λt A, λ1 = λm 1 is a minimal geodesic from A to B with respect to the Thompson metric. Remark 2.13. By considering the extremal cases of α , the maps ⎧ t t t t ⎪ ⎨ λ1 −λm B + λ1 λm −λm λ1 A, λ1 = λm λ1 −λm λ1 −λm φ(t ; A, B) = ⎪ ⎩ λt A, λ1 = λm , 1 and
ψ(t ; A, B) =
⎧ t −t λ1 −λ1 λ11−t −λ1t −1 ⎪ ⎪ −1 B + 1 A, ⎪ ⎪ λ −λ λ1 −λ− 1 1 ⎪ ⎨ −t 1 t t −1 1−t λm −λm λ −λ B + m−1 m A, 1 ⎪ λ− λm −λm ⎪ m −λm ⎪ ⎪ ⎪ ⎩ t λ1 A,
1 λ1 λ− m , λ1 = λm 1 λ1 λ− m , λ1 = λm
λ1 = λm
are minimal geodesics from A to B.
3. New midpoint operations on Pm We have seen in Remark 2.7 that the geometric mean operation (A, B) → A#B is a midpoint operation for any d . In this section we introduce new midpoint operations with respect to the Thompson metric. Theorem 3.1. Let mi min{a, b} for all a, b
: (0, ∞) × (0, ∞) → (0, ∞), i = 1, 2, be maps satisfying
mi (a, b) max{a, b}
> 0 and i = 1, 2. Then the map M : P2m → Pm defined by
M (A, B)
⎧ ⎪ ⎪ ⎪ ⎨√
√
⎪ ⎪ ⎩√
√1
=⎪
λ1 +
1 −1
m1 (λ1
λm +
where λk
(3.4)
,λm ) −1
m2 (λ1 ,λm
)
B+ B+
1 λ1 m1 (λ− 1 , λm )A , 1 λm m2 (λ1 , λ− m )A ,
1 λ− m λ1 1 λ1 λ− m ,
= λk (A−1 B), is a d∞ -midpoint operation on Pm . In particular, the maps
A∗B=
1
√ √ B + λ1 λm A , λ1 + λm
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
AB
⎧ √ ⎪ ⎨ λ1 (A + B), 1+λ
=⎪
√ 1 ⎩ λm (A + B), 1+λm
217
1 λ− m λ1 1 λ1 λ− m ,
are d∞ -midpoint operations on Pm . Furthermore 1 (i) A ∗ B = A B if and only if either λ1 = λm or λ1 = λ− m ; − 1/2 − 1/2 (ii) A#B = A ∗ B if and only if A BA has the spectrum {λ1 , λm }; −1 (iii) A#B = A B if and only if A−1/2 BA−1/2 has the spectrum {λ1 , λ1 }, and in this case
√ √ λ1 λm A#B = A ∗ B = A B = (A + B) = (A + B); 1 + λ1 1 + λm
(iv) A ∗ B = B ∗ A and √ A B = B A; (v) (aA) ∗ (bB)= ab(A ∗ B) for any positive reals a and b;
(vi) A ∗ B
1 2
A+ √
1
λ1 λm
(vii) (MAM ∗ ) ∗ (MBM ∗ ) matrix M.
B and A B
12 (A + B);
= M (A ∗ B)M ∗ and (MAM ∗ ) (MBM ∗ ) = M (A B)M ∗ for any non-singular √
Proof. Let A, B ∈ Pm . Suppose that λ1 = λm . Then B = λ1 A and hence M (A, B) = λ1 A = A#B is −1 1 a d∞ -midpoint between A and B. Suppose that λ1 = λm . If λ− m λ1 , then by (3.4), m1 (λ1 , λm ) ∈
1 −1 [λ− 1 , λm ]. By Remark 2.13, M (A, B) is a d∞ -midpoint between A and B. Similarly for the case λm λ1 . On can see that if m1 (a, b) = max{a, b} and m2 (a, b) = min{a, b}, then M (A, B) = A ∗ B. Similarly if m1 (a, b) = min{a, b} and m2 (a, b) = max{a, b}, then M (A, B) = A B. Therefore A ∗ B and A B
are d∞ -midpoints between A and B. Next, we will check that A ∗ B and A B satisfies (i)–(vii). √ 1 λ1 λm B (i) For λ− m λ1 , A ∗ B = A B if and only if 1 −
1 −1 λ1 = λ− m or λ1 = λm . Similarly for λ1 λm .
(ii) We consider a diagonalization of A−1/2 BA−1/2 D = diag(λ1 , . . . , λm ). Since A#B = A1/2 (A−1/2 BA−1/2 )1/2 A1/2
1/2
√ √ B + λ1 λm A = A λ1 + λm
1/2
=A
D1/2
√
1
1√
λ1 + λm
B+
1
A−1/2 BA−1/2
1
UDU ∗
√ √ λ1 + λm
1/2
=
= UDU ∗ , where U is a unitary matrix and
√ √ λ1 + λm
=A
we have that A#B
√ λ1 λm A if and only if
= A1/2 (UDU ∗ )1/2 A1/2 = A1/2 UD1/2 U ∗ A1/2
and 1
= λ1 1 −
U
1
+ λ1 λm Im
+ λ1 λm Im
√ √ D + λ1 λm Im λ1 + λm
A1/2
A1/2
U ∗ A1/2 ,
√ λ1 λm A if and only if
√ =√ (3.5) D + λ1 λm Im . λ1 + λm √ √ √ √ √ Since x2 − λ1 + λm x + λ1 λm = 0 if and only if x ∈ λ1 , λm , (3.5) is equivalent to λk ∈ {λ1 , λm } for all k = 1, 2, . . . , m. That is, A−1/2 BA−1/2 has the spectrum {λ1 , λm }.
218
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227 1 (iii) Suppose that λ− m
λ1 . We note that λ1 1. Then one can see that A#B = A B if and −1
1 only if A−1/2 BA−1/2 has the spectrum {λ1 , λ1 }. Similarly for λ1 λ− m , A#B = A B if and only if − 1/2 − 1/2 − 1 A BA has the spectrum {λm , λm }. 1 (iv) Let βk = λk (B−1 A), k = 1, 2, . . . , m. Then β1 = λ1 (B−1 A) = λm (A−1 B)−1 = λ− m and
similarly βm
1 = λ− 1 . Therefore,
B∗A=
1
1
1
√ √ (A + (λm λ1 )−1/2 B) A + β1 βm B = −1/2 −1/2 β1 + βm λm + λ1
√ =√ B + λ1 λm A = A ∗ B . λ1 + λm Similarly A B = B A. (v) Let βk = λk ((aA)−1 (bB))
= ba λk (A−1 B) = ba λk . Then
1
√ (aA) ∗ (bB) = √ bB + β1 βm aA β1 + βm =
(b/a)1/2 √
1 √
λ1 +
ab
√
λm
bB + (b/a) λ1 λm aA
√ =√ B + λ1 λm A = λ1 + λm
√
ab(A ∗ B).
√
(vi) It follows from the arithmetic-geometric mean inequality 2 (vii) Straightforward.
λ1 λm λ1 + λm .
Remark 3.2. Classical means on positive reals satisfy (3.4). We note that in Theorem 3.1(ii), A−1/2 BA−1/2 has the spectrum {λ1 , λm } if and only if it has at most two distinct eigenvalues, which is always true for m = 2. Thus, 1 √ A#B = √ B + λ1 λ2 A
λ1 +
λ2
for any 2 × 2 positive definite matrices A and B. An alternate formula of the geometric mean of 2 × 2 positive definite matrices is given by A#B where α 2
=
√ αβ
det(α −1 A + β −1 B)
(α −1 A + β −1 B),
= detA and β 2 = detB. See [5].
4. Convexity of midpoints A complete metric space (M , δ) is called a Hadamard space if it satisfies the semiparallelogram law; for each a, b ∈ M, there exists an m ∈ M satisfying
δ 2 (m, x)
1 2
1
1
2
4
δ 2 (a, x) + δ 2 (b, x) − δ 2 (a, b)
(4.6)
for all x ∈ M. Such spaces are also called (global) CAT(0)-spaces or NPC (non-positively curved) spaces. In particular the m appearing in (4.6) is the unique metric midpoint between a and b. Furthermore there exists a unique minimal geodesic γ : [0, 1] → M between a and b; γ (0) = a, γ (1) = b and for all t ∈ [0, 1],
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
219
δ(γ (t ), γ (s)) = |s − t |δ(γ (0), γ (1)). We note that
δ 2 (x, γa,b (t )) (1 − t )δ 2 (x, a) + t δ 2 (x, b) − t (1 − t )δ 2 (a, b) for all x
(4.7)
∈ M and t ∈ [0, 1] and
δ 2 (a, c ) + δ 2 (b, d) δ 2 (b, c ) + δ 2 (a, d) + δ 2 (a, b) + δ 2 (c , d).
(4.8)
See [19]. The metric space (Pm , d2 ) is a standard example of a NPC space [13,14]. Lemma 4.1. For A, B, X ∈ d2 (X , B) = (1 − t )d2 (A, B).
Pm and t ∈ [0, 1], X = A#t B if and only if d2 (X , A) = td2 (A, B) and
Proof. Let X = A#t B. Then by (2.3), d2 (X , A) td2 (A, B) and d2 (X , B) triangular inequality, the equalities hold true. The converse implication follows from (4.7). We recall that for A, B > 0 and t means of A and B are defined by
(1 − t )d2 (A, B). By the
∈ [0, 1], the t-weighted arithmetic, geometric and harmonic
(1 − t )A + tB, A#t B, ((1 − t )A−1 + tB−1 )−1 , respectively. Definition 4.2. A closed subset ⊂ Pm is said to be geodesically convex (resp. geodesically complete) if A#t B ∈ for all t ∈ [0, 1] (resp. t ∈ R), whenever A, B ∈ . Lemma 4.3. For a closed subset of Pm , is geodesically convex if it is closed under the geometric mean operation. Proof. Suppose that is a closed subset of Pm invariant under the geometric mean operation. Let A, B ∈ and let D = {t ∈ [0, 1] : A#t B ∈ }. Then 0, 1, 1/2 ∈ D. If s, t ∈ D, then by Lemma 2.1(ix), (A#s B)#(A#t B) = A#(s+t )/2 B and thus (s + t )/2 ∈ D. Since D is a closed subset of [0, 1], we have D = [0, 1]. Remark 4.4. If is a non-empty geodesically convex subset of Pm , then a metric projection onto with respect to the Riemannian metric d2 always exists as it does in a Hilbert space (Theorem 6.2.6 [4]). Definition 4.5. A closed subset of Pm is called totally convex if it is invariant under the weighted arithmetic, geometric and harmonic means. For instance, the order interval [A, B] := {X > 0 : A X B} with 0 < A B is totally convex and is a compact subset of P. One can check the total convexity by using the Löwner–Heinz inequality (Lemma 2.1) and the order reverting property of the matrix inversion; 0 < A B implies that B−1 A−1 . Proposition 4.6. Every closed ball with respect to d is geodesically convex. In particular, the closed ball of radius r > 0 around A with respect to the Thompson metric is the order interval [e−r A, er A] and is totally convex. Proof. By (2.3), every closed d -ball is geodesically convex. It follows directly from the definition of the Thompson metric that d∞ (A, e−r A) = r = d∞ (A, er A), and d(A, X ) r for all e−r A X er A. Suppose that d(A, X ) r. Then M (A/X ), M (X /A)
er ,
220
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
so X
er A and A er X, or equivalently e−r A X.
Remark 4.7. It is not easy to describe explicitly the d balls for general , even for d2 . The description of the Thompson balls via order intervals holds true for closed convex cones in a normed space. See Lemma 2.6.2 of [16] for the convexity of Thompson balls. We introduce weighted midpoints for d . Definition 4.8. For t
∈ [0, 1] and A, B ∈ Pm , we denote
M (t ; A, B) = {X > 0 : d (A, X ) = td (A, B), d (B, X ) M (A, B) = M (t ; A, B).
= (1 − t )d (A, B)},
t ∈[0,1]
We note that M (0; A, B) = {A}, M (1; A, B) = {B}. Since the map t geodesic from A to B, A#t B ∈ M (t ; A, B). By the triangular inequality, M (t ; A, B)
→ A#t B is a minimal
= {X > 0 : d (A, X ) td (A, B), d (B, X ) (1 − t )d (A, B)},
the intersection of two balls of radii td (A, B) and (1 − t )d (A, B) around A and B, respectively. Furthermore, M (1/2; A, B) coincides with the set of all d -midpoints of A and B. By Lemma 4.1, M2 (t ; A, B) = {A#t B} for all t ∈ [0, 1]. Theorem 4.9. For A, B convex. Furthermore, (1) (2) (3) (4) (5) (6)
∈ Pm and t ∈ [0, 1], M (t ; A, B) and M (A, B) are compact and geodesically
M (t ; A, A) = {A}; M (t ; A, B) ∩ M (s; A, B) = ∅ if s = t and A = B; M (t ; A, B) = M (1 − t ; B, A); M (t ; A, B)−1 = M (t ; A−1 , B−1 ); M M (t ; A, B)M ∗ = M (t ; MAM ∗ , MBM ∗ ) for all M for any s, t , u ∈ [0, 1], M (t ; A, B)#u M (s; A, B)
∈ GL (m, C);
⊂ M ((1 − u)t + us; A, B).
(4.9)
For the Thompson metric, each M∞ (t ; A, B) is totally convex and M∞ (t ; A, B)
where r
= [r −t A, r t A] ∩ [r t −1 B, r 1−t B],
(4.10)
= max{λ1 (A−1 B), λm (A−1 B)−1 = λ1 (AB−1 )}.
Proof. (1), (2), (3): Straightforward. (4), (5): It follows from the invariancy of the metric under inversion and congruence transformations. (6) Let X ∈ M (t ; A, B) and Y ∈ M (s; A, B). Then d (A, X#u Y ) = d (A#u A, X#u Y )
(2.3)
(1 − u)d (A, X ) + ud (A, Y )
[(1 − u)t + us]d (A, B) and d (B, X#u Y ) = d (B#u B, X#u Y )
(2.3)
(1 − u)d (B, X ) + ud (B, Y )
[(1 − u)(1 − t ) + u(1 − s)]d (A, B). It follows from (1 − u)(1 − t ) + u(1 − s)
= 1 − [(1 − u)t + us] that
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
221
∈ M ((1 − u)t + us; A, B).
X#u Y
Note that M (t ; A, B) is a closed subset of Pm contained in the ball of radius d (A, B) centered at A. By Remark 2.5, each d -ball is compact and hence M (t ; A, B) is compact. By (6), it is geodesically convex. Next, we will show that M (A, B) is compact and geodesically convex. By (6), it is geodesically convex. To show that M (A, B) is compact, it suffices to show that it is a closed subset of Pm from the fact that it is contained in the ball of radius d (A, B) centered at A. Let Xn be a sequence in M (A, B) converging to X ∈ Pm . For each n, pick tn ∈ [0, 1] such that Xn ∈ M (tn ; A, B). Passing to a subsequence, we may assume that tn → t for some t ∈ [0, 1]. Then d (A, X ) = lim d (A, Xn )
= lim tn d (A, B) = td (A, B),
d (B, X ) = lim d (B, Xn )
= lim (1 − tn )d (A, B) = (1 − t )d (A, B)
n→∞ n→∞
n→∞
n→∞
which shows that X ∈ M (A, B). Therefore, M (A, B) is a closed subset of Pm . Finally we deal with the Thompson metric. Let r = max{λ1 (A−1 B), λm (A−1 B)−1 = λ1 (AB−1 )}. Then d∞ (A, B) = d∞ (I , A−1/2 BA−1/2 ) = log r. By the fact that M∞ (t ; A, B) is the intersection of the balls of radii t log r and (1 − t ) log r around A and B and by Proposition 4.6, (4.10) holds. This implies in particular that M∞ (t ; A, B) is totally convex. Remark 4.10. From M∞ (t ; A, B)
⊂ [r −t A, r t A] ∩ [r t −1 B, r 1−t B] ⊂ [(1/r )A, rA] ∩ [(1/r )B, rB]
for all t ∈ [0, 1], we have M∞ (A, B) ⊂ [(1/r )A, rA] λ1 (AB−1 )}. In general, M∞ (A, B) is not totally convex. Corollary 4.11. Let A, B matrix.
∩ [(1/r )B, rB] where r = max{λ1 (A−1 B),
∈ Pm . Let A−1/2 BA−1/2 = UDU ∗ , where U is a unitary matrix and D is a diagonal
(1) M (t ; A, B) = A1/2 U ∗ M (t ; Im , D)UA1/2 and M (A, B) = A1/2 U ∗ M (Im , D)UA1/2 . (2) Let t0 ∈ (0, 1). Then ⎧ t ⎪ ⎨ A1/2 M (t0 ; Im , A−1/2 BA−1/2 ) t0 A1/2 , t t0 M (t ; A, B) ⊂ 1−t ⎪ ⎩ B1/2 M (t0 ; B−1/2 AB−1/2 , Im ) 1−t0 B1/2 , t t0 .
(4.11)
In particular, M (t ; Im , D)
⊂
⎧ 1 ⎨M ;I
⎩ D#
2
m, D
2(1−t ) M
2t
,
1 ;I ,D 2 m
,
t
t
1 2 1 . 2
(4.12)
(3) Let D be a diagonal matrix with positive entries. If M (t0 ; I , D) consists of diagonal matrices for some t0 ∈ (0, 1), then M (I , D) consists of diagonal matrices. (4) If M (t0 ; A, B) = {A#t0 B} for some t0 ∈ (0, 1), then the curve t → A#t B is a unique minimal geodesic with respect to d from A to B. Proof. (1) It follows from Theorem 4.9. (2) Let t0 ∈ (0, 1) and let t t0 . Let X B# 1−t0 X 1−t
Since B#u X
∈ M (t ; A, B). Then by Theorem 4.9(6),
∈ M (1; A, B)# 1−t0 M (t ; A, B) ⊂ M (t0 ; A, B).
=B
1−t
1/2
(B−1/2 XB−1/2 )u B1/2 , we have
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Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
X
1−t
1−t
∈ B1/2 [B−1/2 M (t0 ; A, B)B−1/2 ] 1−t0 B1/2 = B1/2 M (t0 ; B−1/2 AB−1/2 , Im ) 1−t0 B1/2 . 1−t
Therefore M (t ; A, B) ⊂ B1/2 M (t0 ; B−1/2 AB−1/2 , Im ) 1−t0 B1/2 . Suppose that t t0 . Then 1 − t 1 − t0 and by the preceding step, M (t ; A, B) = M (1 − t ; B, A) t
⊂ A1/2 M (1 − t0 ; A−1/2 BA−1/2 , Im ) t0 A1/2 t
= A1/2 M (t0 ; Im , A−1/2 BA−1/2 ) t0 A1/2 . Next, we will show (4.12) for t M (t ; Im , D) ⊂ D
1/2
12 . By (4.11),
M (1/2; D
−1
, Im )2(1−t ) D1/2
= D1/2 (Im #2(1−t ) M (1/2; D−1 , Im ))D1/2 = D#2(1−t ) M (1/2; Im , D)). (3) It follows from (4.11). (4) Suppose that M (t0 ; A, B) M (t0 ; A, B) = A
1/2
= {A#t0 B} for some t0 ∈ (0, 1). By (1), −1/2
M (t0 ; Im , A
BA−1/2 )A1/2
= {A#t0 B}
= A1/2 {(A−1/2 BA−1/2 )t0 }A1/2 which shows that M (t0 ; I , A−1/2 BA−1/2 ) M (t0 ; A, B) = M (1 − t0 ; B, A)
= {(A−1/2 BA−1/2 )t0 }. Similarly
= B1/2 M (1 − t0 ; Im , B−1/2 AB−1/2 )B1/2
= {A#t0 B} = {B#1−t0 A} = B1/2 {(B−1/2 AB−1/2 )1−t0 }B1/2 yields that M (t0 ; B
−1/2
AB−1/2 , Im )
= M (1 − t0 ; Im , B−1/2 AB−1/2 ) = {(B−1/2 AB−1/2 )1−t0 }.
By (4.11), the proof is completed. 5. Uniqueness of d∞ -minimal geodesic In this section we find a uniqueness criterion for minimal geodesics with respect to the Thompson metric. We denote σ (A) by the spectrum of A. We note that for Hermitian matrices A and B, A is similar to B if and only if A is unitarily similar to B if and only if λi (A) = λi (B) for all 1 i m. We denote A ∼ B if and only if A is similar to B. Proposition 5.1. Let A, B ∈ Pm such that σ (A) ⊆ {a, b} for some a (unitarily) similar to A1/2 and A−1/2 respectively, then B = A1/2 .
√
b > 0. If B and A−1/2 BA−1/2 are
Proof. Suppose that a = b. Then A = aIm and hence B = aIm = A1/2 . Suppose that σ (A) = {a, b} where a > b > 0. Pick a unitary matrix U such that A = UDU ∗ where D is a diagonal matrix. Setting X = U ∗ BU, X is unitarily similar to D1/2 and D−1/2 XD−1/2 is unitarily similar to D−1/2 . Indeed, X ∼ B ∼ A1/2 = UD1/2 U ∗ ∼ D1/2 and D−1/2 XD−1/2 = (U ∗ A−1/2 U )(U ∗ BU )(U ∗ A−1/2 U ) ∼ A−1/2 BA−1/2 ∼ A−1/2 ∼ D−1/2 . If X = D1/2 , then B = UXU ∗ = UD1/2 U ∗ = A1/2 . So, it is enough to show that X = D1/2 . By assumption, D has the spectrum {a, b}.
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
223
Pick a permutation matrix P such that
= PDP = diag(a, . . . , a, b, . . . , b),
D1
m−k
k
1/2
= PXP, we have X1 ∼ PD1/2 P = D1 and D1−1 X1 = 1/2 1/2 . If X1 = D1 , then X = PX1 P = PD1 P = D1/2 . Therefore,
where k denotes the number of a’s in D. Setting X1 PD−1 PPXP
PD−1 XP
=
∼D
−1/2
1/2
∼ D1
we may assume that D
= diag(a, . . . , a, b, . . . , b).
k
m−k
Write X = [xij ]m×m ∈ Pm . Let λk = λk (X ) and βk = λk (D D−1/2 XD−1/2 are similar to D1/2 and D−1/2 respectively,
λi =
√
(1 i k),
a 1
βi = √
λi =
= λk (D−1/2 XD−1/2 ). Since X and
(k < i m),
b
1
βi = √
(1 i m − k),
b
√
−1 X )
(m − k < i m).
a
By Hadamard’s inequality (Theorem 7.8.1 [10]), √ k √ m−k a b
=
m i=1
λi = det(X )
m i=1
xii .
(5.13)
By the spectral theorem for normal matrices (Theorem 2.5.4 of [10]), ka + (m − k)b =
m i=1
+2
xii2
1i
m
i=1
λ2i =
m
i,j=1
|xij |2 . That is,
|xij |2 .
(5.14)
We note that the right-hand side is equal to m i=1
xii2
+
|xij |2 +
(i,j)∈I11
|xij |2 +
(i,j)∈I12
|xij |2 +
(i,j)∈I21
|xij |2 ,
(i,j)∈I22
where I11
= {(i, j) : 1 i, j k, i = j}, I12 = {(i, j) : 1 i k, k < j m},
I21
= {(i, j) : k < i m, 1 j k, }, I22 = {(i, j) : k < i, j m, i = j}.
By the same argument for D−1 X, we have m−k b
+
k a
=
m i=1
+
βi2 =
1
k 1
a2
b2 (i,j)∈I 21
i=1
xii2
+
|yij |2 +
1 b2
1
m i=k+1
b2 (i,j)∈I 22
xii2
+
1 a2
|yij |2 +
(i,j)∈I11
1 a2
|yij |2
(i,j)∈I12
|yij |2 .
Combining this together with (5.14) yields m−k b
(a − b ) = (a − b ) 2
2
2
2
m i=k+1
⎛ xii2
+ (a − b ) ⎝ 2
2
(i,j)∈I21
|xij | + 2
(i,j)∈I22
⎞ 2⎠
|xij |
.
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Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
In particular, implies that m
m
(m − k)b. This together with the arithmetic-geometric mean inequality
2 i=k+1 xii
m xii
i=k+1
m−k 2
2 i=k+1 xii
m−k
√ m−k b .
(5.15)
By (5.13) and (5.15), √ k √ m−k a b
m
xii
i=1
=
k
xii
i=1
·
m
xii
i=k+1
k √ m−k b · xii . i=1
That is, k
xii
i=1
√ k a .
(5.16)
Now, we compute the trace of X and D−1 X; k
√
√
a + (m − k)
b
= Tr(X ) =
m
xii
(5.17)
i=1
and m−k
√
k
+ √ = Tr(D−1 X ) = a
b
k 1
a i=1
xii
+
m 1
b i=k+1
xii .
Combining these two equations yields m
xii
i=k+1
√ = (m − k) b
(5.18)
and hence from (5.17), k i=1
xii
√ = k a.
(5.19)
By (5.16) and (5.19), √ k a
k
xii
i=1
=
i=1 xii
=
k
=
√ k a
√
√
which implies that xii = |xij |2 = 0. 1i
That is, X
k
a for all i = 1, . . . , k. This with (5.13) and (5.18) yields m m−k m √ m−k i=k+1 xii xii = b m−k i=k+1
which implies that xii √ m−k b
k
= D1/2 .
b for k
< i m. By (5.14) we conclude that
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
225
The following is one of the main results of this paper. Theorem 5.2. Let A, B ∈ Pm . Then there is a unique d∞ -minimal geodesic from A to B if and only if there is a unique midpoint between A and B if and only if σ (A−1 B) ⊆ {a, a−1 } for some a > 0. Proof. Let A, B
∈ Pm and let λk = λk (A−1 B), k = 1, 2, . . . , m. Then it suffices to show the following.
(i) If λ1 = λm , then there is a unique d∞ -minimal geodesic (midpoint) from A to B. (ii) If A−1 B has at least three distinct eigenvalues, then there are infinitely many d∞ -minimal geodesics (midpoints) from A to B. 1 (iii) If A−1 B has two distinct eigenvalues with λ1 = λ− m , then there are infinitely many d∞ -minimal geodesics (midpoints) from A to B. 1 (iv) If A−1 B has two distinct eigenvalues with λ1 = λ− m , then there is a unique d∞ -minimal geodesic (midpoint) from A to B.
= λ1 = λm . Then B = aA. By Corollary 4.11, it suffices to show that √ M∞ (1/2; Im , aIm ) = aIm .
(i) Let a
We further assume that a > 1 because M∞ (1/2; Im , aIm ) = M∞ (1/2; Im , a−1 Im )−1 . Then d∞ (Im , aIm ) = log a. Let X ∈ M∞ (1/2; Im , aIm ) and let λk = λk (X ). 1 Suppose that λ1 λ− m . Then 1 2 = d∞ (Im , aIm ) = 2d∞ (Im , X ) = 2 max{log λ1 , λ− m } = log λ1 √ implies that λ1 = a. Since λ1 (a−1 X ) = a−1 λ1 a−1 λm = λm (a−1 X ), we have that
log a
log
√
1
d∞ (Im , aIm ) = d∞ (aIm , X ) = d∞ (Im , a−1 X ) ! " a a = log = max log a−1 λ1 , log
a=
2
λm λm √ which yields λm = a. That is, X = aIm . √ 1 1 a = d∞ (Im , X ) = log λ− Suppose that λ1 < λ− m . Then log m implies that λm = √
√1 . But this is a
impossible from the fact that log
√
a
= d∞ (aI , X ) log λm (a−1 X )−1 = log
a
λm
√ = log a a.
(ii) By Remark 2.13 and Theorem 3.1(iii), the map
φ(t ; A, B) =
λt1 − λtm λ1 λtm − λm λt1 B+ A λ1 − λm λ1 − λm
is a minimal geodesic for A to B with φ(1/2; A, B) = A#B. By Proposition 2.9, there are infinitely many minimal geodesics and midpoints from A to B. −1 1 (iii) By Theorem 2.12, if λ− m < λ1 , then for any α ∈ (λ1 , λm ), fα (t ) =
λt1 − α t λ1 α t − αλt1 B+ A λ1 − α λ1 − α
is a minimal geodesics with respect to the Thompson metric from A to B. Also, if λ1 1 any α ∈ (λ1 , λ− m ), gα (t ) =
α t − λtm αλtm − λm α t B+ A α − λm α − λm
1 < λ− m , then for
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Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
is a minimal geodesics with respect to the Thompson metric from A to B. Then one can see that A#B = fα (1/2) and A#B = gα (1/2). Therefore, there are infinitely many minimal geodesics and midpoints from A to B. (iv) We will show that {A#B} = M(1/2; A, B). By Corollary 4.11, we may assume that A = Im and B
= D = diag(a, . . . , a, a−1 , . . . , a−1 ), a > 1.
m−k
k
Let X ∈ M(1/2; Im , D). We claim that X = D1/2 . By Proposition 5.1, it is enough to show that X and D−1/2 XD−1/2 are similar to D1/2 and D−1/2 , respectively. Let λk = λk (X ) and βk = λk (D−1 X ) = λk (D−1/2 XD−1/2 ). From d∞ (I , X ) = d∞ (X , D)(= d∞ (Im , D−1/2 XD−1/2 )) = 12 log a, we have 1 √ λi , βi ∈ √ , a a for all 1 i m. Applying the inequalities λi (A)λm (B) A, B ∈ Pm (Theorem 7.10 [23]), we have $ $ # #
λm a
λ1
βm min
#
a
, aλm max
max aλm ,
$
λi
λ1 a
λi (AB) = λi (BA) λi (A)λ1 (B) for any
, aλm β1 aλ1 ,
(5.20)
βi aλi (1 i m − k),
a
#
λi
βi min aλi ,
a
λ1
(5.21)
$
a
(i > m − k).
(5.22)
By Lidskii’s theorem (Corollary III.4.6 [2]), m −k i=1 m i=1
λi (D−1 )
m i=k+1
m −k i=1
βi ,
βi = det(D−1 X ) = am−2k
We note that βi
=
from (5.21) and λm am−k
m i=k+1
or
λi
%m
i=k+1 λi
√1 for all i
λi
√1
a √1 . a
(5.23)
m i=1
λi .
(5.24)
> m − k by (5.22) and λ1
√
a. Also, βi
=
√
a for all 1
By (5.23),
√ m−k a ,
m−k
a
. Since λi
√1 for all i a
= 1, 2 . . . , m, we have
1
λi = √ , i > m − k . a
However, by (5.24) √ m−2k a
=
√ m−k √ −k a a
= am−2k
=
m i=1
k √ k−m a λi i=1
βi = am−2k
m i=1
λi = am−2k
k i=1
λi
m i=k+1
λi
i m−k
Y. Lim / Linear Algebra and its Applications 439 (2013) 211–227
implies that
λi =
%k
√
i=1
a, i
λi =
√ k a . Since λi
227
√
a for all i, we have
= 1, 2, . . . , k.
This shows that λi (X )
= λi (D1/2 ) and λi (D−1 X ) = λi (D−1/2 ) for all i = 1, . . . , m.
Acknowledgement The author wishes to express his gratitude to Professor Bas Lemmens for helpful insights about convexity of Thompson balls. This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (No. 2012-005191). References [1] F. Barbaresco, Innovative tools for radar signal processing based on Cartan’s geometry of symmetric positive definite matrices and information geometry, in: Proceedings of the IEEE International Radar Conference, Rome, Italy, 2008. [2] R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics,, Springer, 1996, pp. . [3] R. Bhatia, On the exponential metric increasing property, Linear Algebra Appl. 375 (2003) 211–220. [4] R. Bhatia, J. Holbrook, Riemannian geometry and matrix geometric means, Linear Algebra Appl. 413 (2006) 594–618. [5] R. Bhatia, Positive Definite Matrices, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2007. [6] M. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, Berlin, 1999. [7] H. Busemann, B.B. Phadke, Spaces with distinguished geodesics, Monographs and Textbooks in Pure and Applied Mathematics, vol. 108, Marcel Dekker Inc., New York, 1987. [8] G. Corach, H. Porta, L. Recht, Convexity of the geodesic distance on spaces of positive operators, Illinois J. Math. 38 (1994) 87–94. [9] P.D. Fletcher, S. Joshi, Riemannian geometry for the statistical analysis of difusion tensor data, Signal Process. 87 (2007) 25–262. [10] R. Horn, C. Johnson, Matrix Analysis, Cambridge University Press, 1985. [11] U. Itai, N. Sharon, Subdivision schemes for positive definite matrices, Found. Comput. Math. http://dx.doi.org/10.1007/ s10208-012-9131-y. [12] F. Kubo, T. Ando, Means of positive linear operators, Math. Ann. 246 (1980) 205–224. [13] S. Lang, Fundamentals of Differential Geometry, Graduate Texts in Mathematics, Springer-Verlag, 1999. [14] J.D. Lawson, Y. Lim, The geometric mean, matrices, metrics, and more, Am. Math. Mon. 108 (2001) 797–812. [15] J. Lawson, Y. Lim, Metric convexity of symmetric cones, Osaka J. Math. 44 (2007) 795–816. [16] B. Lemmens, R. Nussbaum, Nonlinear Perron–Frobenius Theorey, Cambridge Tracts in Mathematics, vol. 189, Cambridge University Press, 2012. [17] X. Pennec, P. Fillard, N. Ayache, A Riemannian framework for tensor computing, Int. J. Comput. Vision 66 (2006) 41–66. [18] R.D. Nussbaum, Finsler structures for the part metric and Hilbert’s projective metric and applications to ordinary differential equations, Differ. Integral Equ. 7 (1994) 1649–1707. [19] K.-T. Sturm, Probability measures on metric spaces of nonpositive curvature, in: P. Auscher, et al (Ed.), Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Contemporary Mathematics, vol. 338, American Mathematical Society (AMS), Providence, 2003. [20] A.C. Thompson, On certain contraction mappings in a partially ordered vector space, Proc. Am. Math. Soc. 14 (1963) 438–443. [21] M. Vesentini, Invariant metrics on convex cones, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3 (1976) 671–696. [22] M. Zerai, M. Moakher, The Riemannian geometry of the space of positive-definite matrices and its application to the regularization of positive-definite matrix-valued date, J. Math. Imaging Vision 40 (2011) 171–187. [23] F. Zhang, Matrix Theory: Basic Results and Techniques, Universitext, Springer-Verlag, New York, 1999.