Direct sums of positive semi-definite matrices

Linear Algebra and its Applications 463 (2014) 273–281

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Linear Algebra and its Applications www.elsevier.com/locate/laa

Direct sums of positive semi-definite matrices Jean-Christophe Bourin a,1 , Eun-Young Lee b,∗,2 a

Laboratoire de mathématiques, Université de Franche–Comté, 25 000 Besançon, France b Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea

a r t i c l e

i n f o

Article history: Received 20 June 2014 Accepted 6 September 2014 Available online 20 September 2014 Submitted by R. Bhatia MSC: 15A24 15A60

a b s t r a c t We show a remarkable link between the direct sum and the usual sum of m positive semi-definite matrices. The direct sum is a kind of average of m copies of the usual sum. This averaging is applied to obtain several inequalities related to the Rotfel’d majorizations for convex/concave functions. © 2014 Elsevier Inc. All rights reserved.

Keywords: Positive definite matrices Partitioned matrices Majorization Eigenvalue inequalities Convex/concave functions

1. Direct sums are averages of sums There exists a useful relation between the direct sum A ⊕ B and the usual sum A + B of two positive semi-definite n × n matrices, written as * Corresponding author. E-mail addresses: [email protected] (J.-C. Bourin), [email protected] (E.-Y. Lee). Partially supported by ANR 2011-BS01-008-01. 2 Research supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2013-R1A1A2059872). 1

http://dx.doi.org/10.1016/j.laa.2014.09.012 0024-3795/© 2014 Elsevier Inc. All rights reserved.

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A⊕B ≺A+B

(1.1)

and stated as A ⊕ B is majorized by A + B. This says that, for all k = 1, . . . , 2n, k  j=1

λj [A ⊕ B] ≤

k 

λj [A + B]

j=1

with equality for k = 2n. Here, given a Hermitian m × m matrix Z, we use the notation λj [Z] for the eigenvalues of Z arranged in decreasing order, 1 ≤ j ≤ m, and we merely set λj [Z] := 0 for j > m. Majorization is a flexible tool in deriving matrix inequalities. For instance, the above majorization implies that, given a convex function g(t) such that g(0) ≤ 0, one has k  j=1

k      λj g(A ⊕ B) ≤ λj g(A + B)

(1.2)

j=1

for all 1 ≤ k ≤ 2n. This can be written as a so-called weak majorization, g(A ⊕ B) ≺w g(A + B).

(1.3)

This weak majorization was established by Rotfel’d [10] in 1969. He used it to show a famous trace inequality, Tr g(A) + Tr g(B) ≤ Tr g(A + B), corresponding to the case k = 2n in (1.2). Of course for a concave function f (t) such that f (0) ≥ 0, the inequalities (1.2) are reversed, and this is written as a so-called supermajorization, f (A ⊕ B) ≺w f (A + B),

(1.4)

containing in particular the Rotfel’d trace inequality for concave functions, Tr f (A) + Tr f (B) ≥ Tr f (A + B). Since such inequalities for convex/concave functions are derived from the majorization between A +B and A ⊕B, we may expect that a new insight on direct and standard sums will provide further inequalities completing (1.3)–(1.4). The main result of this note is a nice relation for these two type of sums. Let M+ n be the cone of positive semi-definite n × n matrices. A matrix V ∈ Mm,n , the space of m × n matrices, is called an isometry whenever V ∗ V is the identity of Mn . Denote Im := {1, . . . , m}.

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275

Theorem 1.1. Let Ai ∈ M+ n , i ∈ Im . Then, for some isometries Vk ∈ Mmn,n , k ∈ Im , m 

1  Ai = Vk m i=1 m



m 

 Ai Vk∗ .

i=1

k=1

This says that direct sums are averages of usual sums, up to isometric congruences. It is a genuine non-commutative fact with no analogous statement for positive vectors and permutations of their components. The proof will be given in Section 3. The next section deals with several applications of the theorem improving or completing the Rotfel’d majorizations. 2. Applications to convex and concave functions Theorem 1.1 may be used to improve the Rotfel’d majorizations (1.3)–(1.4). We also need the following special case of [2, Corollary 2.4]. Theorem 2.1. Let Sk ∈ M+ n , k ∈ Im , and let f (t) be a concave function on [0, ∞). Then, for some unitaries U, V ∈ Mn ,  f

1  Sk m m



k=1

    m m 1  1 1  ∗ U ≥ f (Sk ) U + V f (Sk ) V ∗ . 2 m m k=1

k=1

If furthermore f (t) is monotone, then we can take U = V . Combined with Theorem 1.1 we may obtain the following corollary. Corollary 2.2. Let Ai ∈ M+ n , i ∈ Im . Let f (t) be a concave function on [0, ∞) such that f (0) ≥ 0. Set ν = 2 if f (t) is not monotone, and set ν = 1 if f (t) is monotone. Then, for some isometries Wk ∈ Mmn,n , k ∈ Iνm , f

 m 

Ai

i=1

Proof. Let S :=

m i=1

  m  νm  1  ≥ Wk f Ai Wk∗ . νm i=1 k=1

Ai . Theorem 1.1 says that m 

1  Vk SVk∗ m m

Ai =

i=1

k=1

for some isometries Vk ∈ Mmn,n , k ∈ Im . Hence,  f

m  i=1

Ai

 =f

m 1  ∗ Vk SVk . m k=1

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Applying Theorem 2.1 with Sk = Vk SVk∗ gives f

m 

Ai

i=1

    m m 1   1 1   ∗ ∗ ∗ U ≥ U +V V∗ f Vk SVk f Vk SVk 2 m m k=1

k=1

for some unitaries U, V ∈ Mmn , with U = V if f (t) is monotone. Let Ek ∈ Mmn be the orthoprojection onto the kernel of V ∗ and observe that  f Vk SVk∗ = V f (S)Vk∗ + f (0)Ek . Since f (0) ≥ 0, the previous inequality entails f

m 

Ai

i=1

    m m 1  1 1  ∗ ∗ ∗ U ≥ Vk f (S)Vk U + V Vk f (S)Vk V ∗ . 2 m m k=1

k=1

Setting Wk = U Vk for 1 ≤ k ≤ m and Wk = V Vk for m + 1 ≤ k ≤ 2m completes the proof. 2 Of course, letting g(t) = −f (t), the inequality of Corollary 2.2 is reversed for convex functions g(t) such that g(0) ≤ 0. In the simple case of a monotone convex function and two operators we have a significant improvement of (1.3). Corollary 2.3. Let A, B ∈ M+ n and let g(t) be a monotone convex function on [0, ∞) such that g(0) ≤ 0. Then, for some isometries U, V ∈ M2n,n , g(A ⊕ B) ≤

 1 U g(A + B)U ∗ + V g(A + B)V ∗ . 2

The next series of corollaries nicely complements the eigenvalue inequalities contained in the Rotfel’d majorizations (1.3)–(1.4). Denote I− n := {0, 1, . . . , n − 1}. Corollary 2.4. Let A, B ∈ M+ n and let g(t) be an increasing convex function on [0, ∞). Then   1     λj+1 g(A + B) + λk+1 g(A + B) λj+k+1 g(A ⊕ B) ≤ 2 for all j, k ∈ I− n. Proof. If the corollary holds for the function g(t), then it also holds for the function g(t) + c for any constant c. Hence we may suppose g(0) = 0 and so g(t) is nonnegative. Corollary 2.3 implies   1   λj+k+1 g(A ⊕ B) ≤ λj+k+1 U g(A + B)U ∗ + V g(A + B)V ∗ . 2

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This, combined with Weyl’s inequality, λj+k+1 [S + T ] ≤ λj+1 [S] + λk+1 [T ]

(2.1)

for all Hermitian S, T ∈ M2n and all j, k ∈ I− 2n such that j + k + 1 ≤ 2n, proves ∗ the corollary by taking S = U g(A + B)U , T = V g(A + B)V ∗ and by noting that λl+1 [U g(A + B)U ∗ ] = λl+1 [V g(A + B)V ∗ ] = λl+1 [g(A + B)] whenever l ∈ I− n thanks to the positivity of g(A + B). 2 Corollary 2.5. Let A, B ∈ M+ n and let f (t) be an increasing concave function on [0, ∞). Then   1     λj+k+1 f (A + B) + λn−k f (A + B) λj+1 f (A ⊕ B) ≥ 2 for all j, k ∈ I− n such that j + k + 1 ≤ n. Proof. The proof is similar to the previous one by using the following version of (2.1), λj+1 [S + T ] ≥ λj+l+1 [S] + λ2n−l [T ]

(2.2)

for all Hermitian S, T ∈ M2n and all j, k ∈ I− 2n such that j + k + 1 ≤ 2n. Taking − j ∈ I− , l = k + n with k ∈ I and using the version of Corollary 2.3 for a concave n n function f (t) = −g(t) (we may suppose f (t) ≥ 0), we obtain the desired inequality by setting in (2.2) S = U f (A + B)U ∗ and T = V f (A + B)V ∗ and by noting (again) that λl+1 [U f (A + B)U ∗ ] = λl+1 [V f (A + B)V ∗ ] = λl+1 [f (A + B)] whenever l ∈ I− n. 2 For a Hermitian matrix Z ∈ Mm , we have −λp [Z] = λm−p+1 [−Z] for all p ∈ Im . Using this, we see that Corollary 2.5 provides some eigenvalue inequalities for decreasing convex functions −f (t), meanwhile Corollary 2.4 entails some eigenvalue inequalities for decreasing concave functions. Corollaries 2.4–2.5 in case of j = k have the following form. Corollary 2.6. Let A, B ∈ M+ n and let f (t) be a decreasing concave function on [0, ∞). Then     λ2n−2j f (A ⊕ B) ≥ λn−j f (A + B) for all j ∈ I− n. Corollary 2.7. Let A, B ∈ M+ n and let g(t) be a decreasing convex function on [0, ∞). Then   1     λn−2j g(A + B) + λj+1 g(A + B) λ2n−j g(A ⊕ B) ≤ 2 for all j ∈ I− n such that 2j + 1 ≤ n.

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These six corollaries may obviously be stated for functions on (0, ∞) if A, B are positive definite. To illustrate Corollary 2.7 in this context, pick g(t) = t−1 and two invertible matrices A, B ∈ M+ 7 . Then, with j = 2, we have     λ12 A−1 ⊕ B −1 ≤ λ3 (A + B)−1 . Of course such an inequality cannot be extended to all decreasing functions on (0, ∞). 3. Comments and proof of Theorem 1.1 For nonnegative scalars ai , i ∈ Im , we have the trivial inequality a1 + · · · + am ≤ max{a1 , · · · , an }. m By the simple fact that λj [A] ≥ λj [B] whenever A ≥ B, deleting m − 1 terms in the sum of Theorem 1.1 we obtain a matrix version of this inequality. Let Ai ∈ M+ n , i ∈ Im . Then, for all j ∈ In ,    m  m  1  λj A i ≤ λj Ai . m i=1 i=1 This is an elementary observation since the matrix in the left hand side can be written m as C ∗ ( i=1 Ai )C where C ∈ Mn,1 is the contractive mapping with all entries equal √ to 1/ m. A subtler consequence of Theorem 1.1 can be obtained via Weyl’s inequality (2.1). This is Corollary 2.4 with g(t) = t and a direct sum of m matrices in lieu of a pair. − Corollary 3.1. Fix a positive integer m. Let Ai ∈ M+ n , i ∈ Im , and let jk ∈ In , k ∈ Im .

m Suppose that j := k=1 jk ≤ mn − 1. Then

 λj+1

m  i=1

 Ai

 m  m  1  ≤ λjk +1 Ai . m i=1 k=1

Let A ∈ M+ n and set aj := λj [A], 1 ≤ j ≤ n. Then A is unitarily equivalent to the diagonal matrix a1 ⊕ · · · ⊕ an

(3.1)

so that Theorem 1.1 yields the following simple statement which is related to a theorem of Fillmore. The symbol τ stands for the normalized trace of A. Corollary 3.2. Let A ∈ M+ n . Then, for some rank one orthoprojections Ej , j ∈ In , A=τ

n  j=1

Ej .

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Suppose now that the ai ’s in (3.1) sum up to an integer m ≥ n. By subtracting to each ai , some integer ni ≤ ai , we may decompose A = B + C where B has a spectrum lying on N and Tr C = rank(C). From this and the above corollary we easily infer Fillmore’s theorem [6]: Corollary 3.3. Let A ∈ M+ n and suppose Tr A = m is an integer. Then A is a sum of orthoprojections if and only if m ≥ rank(A). Theorem 1.1 can be regarded as a far extension of Corollary 3.2. We turn to its proof. Proof of Theorem 1.1. We will derive Theorem 1.1 from a useful decomposition, [2, Lemma 3.4] (see also [7, Chapter II, Theorem 2.5]): For every matrix in M+ n+m partitioned in blocks, we have 

A X∗



X B

 =U

A 0

   0 0 0 U∗ + V V∗ 0 0 B

(3.2)

for some unitaries U, V ∈ Mn+m . To obtain this nice decomposition, factorize the blockmatrix as a square of positive matrices, 

A X∗

X B



 =

C Y∗

Y D



C Y∗

Y D



and observe that it can be written as 

C Y∗

0 0



C 0

Y 0



 +

0 Y 0 D



0 Y∗

0 D



= T ∗ T + S ∗ S.

The polar decompositions of T and S show that T ∗ T and S ∗ S are unitarily congruent to 

A TT = 0 ∗

0 0



 0 0 and SS = . 0 B ∗



This gives (3.2). Now, let m be a positive integer and let Ai ∈ M+ n , i ∈ Im . Define a matrix W = [Wk,l ] ∈ Mm [Mn ] partitioned in m × m blocks Wk,l ∈ Mn by setting for all k, l ∈ Im 1 Wk,l = √ ω kl I m where ω = ei2π/m is the primitive m-root of 1 and I stands for the n × n identity matrix. Observe that W is a unitary matrix that enjoys the property that the partitioned matrix Z ∈ Mm [Mn ],

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m 

Z := W

Ai W ∗

(3.3)

i=1

has all its diagonal blocks Zj,j equal to 1  Ai . m i=1 m

By an obvious version of (3.2) for positive matrices partitioned in m × m blocks, we infer that 1  Z= Uj m j=1 m



 m 1  Ai Uj∗ m i=1

for some isometries Uj ∈ Mmn,n . This combined with (3.3) yields the relation of the theorem by letting Vj := W ∗ Uj . 2 Remark 3.4. A look at the proof of [9, Theorem 2] shows that the decomposition (3.2) is implicitly used in a paper by Nielsen on quantum physics to prove that, in case of n = m, the diagonal blocks satisfy the majorization A + B ≺ A↓ + B ↓ . Here X ↓ stands for the diagonal matrices with the eigenvalues of X arranged in decreasing. The idea is credited to Wielandt. The explicit form of (3.2) with unitaries U , V is given in [2] with some related matrix inequalities. Such a form involving unitary orbits is a significant fact opening the way to several striking further decompositions of positive matrices partitioned in Hermitian blocks, see [3] and [5]. Theorem 1.1 is clearly one more application of this unitary orbit technique. Remark 3.5. In case of m = 2p for some integer p > 0, then we may replace the matrix W in the above proof by a matrix with real entries as shown in the proof of [3, Theorem 2.1]. Hence for m = 2p , and positive semi-definite matrices Ai with real entries, Theorem 1.1 can be stated with matrices Ui with real entries too. Remark 3.6. In case of m = 2, i.e., the averaging relation between positive matrices A ⊕ B and A + B, Theorem 1.1 is rather well-known folklore, see the proof of [1, Theorem IV.2.13]. This proof, which does not use (3.2), can inductively cover the case m = 2p . For a general integer m, the decomposition (3.2) seems to be a crucial step. Remark 3.7. The decomposition (3.2) is used in the twin papers [4–8] to give some extension to Fillmore’s theorem in the setting of operators on an infinite dimensional Hilbert space. Theorem 1.1 still holds for operators on an infinite dimensional Hilbert space.

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