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Reduction of joint c-numerical ranges
Applied Mathematics and Computation 232 (2014) 178–182
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Reduction of joint c-numerical ranges M.T. Chien a,⇑,1, H. Nakazato a,b,2 a b
Department of Mathematics, Soochow University, Taipei 11102, Taiwan Department of Mathematical Sciences, Faculty of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan
a r t i c l e
i n f o
a b s t r a c t Let ðH1 ; H2 ; . . . ; Hm Þ be an m-tuple of n-by-n Hermitian matrices and c 2 Rn . We investigate the reduction problem of the joint c-numerical range W c ðH1 ; H2 ; . . . ; Hm Þ. In particular, for m ¼ 3 and n P 3, we prove that W c ðH1 ; H2 ; H3 Þ is attainable by the joint numerical range WðK 1 ; K 2 ; K 3 Þ for some triple Hermitian matrices ðK 1 ; K 2 ; K 3 Þ. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Joint c-numerical range Reduction Compound matrix Robust control
1. Introduction Let A 2 M n , the algebra of n-by-n complex matrices. For a real n-tuple c ¼ ðc1 ; c2 ; . . . ; cn Þ, the c-numerical range W c ðAÞ of A is defined as the set
( ) n X n W c ðAÞ ¼ cj xj Axj : fx1 ; x2 ; . . . ; xn g is an orthonormal basis for C : j¼1
Westwick [18] proved the convexity of the range W c ðAÞ. If c ¼ ð1; . . . ; 1; 0; . . . ; 0Þ, with 1’s appearing k times, W c ðAÞ becomes the k-numerical range W k ðAÞ. In particular, k ¼ 1; W k ðAÞ reduces to the classical numerical range which is usually denoted by WðAÞ. The subject of various numerical ranges has been extensively developed and studied (see, for instance, [8,13]), and has connections to many different branches of science, such as quantum mechanics [7] and stability theory [6,14]. It is shown in [3] that for some nilpotent Toeplitz matrix A 2 M n and any c 2 Rn , there exists an n!-by-n! matrix B satisfying W c ðAÞ ¼ WðBÞ. Furthermore, the paper [4] proves that this reduction property is also true for arbitrary matrix A 2 M n , and the size of B can be decreased to
n
p1 p2 p‘
¼
n! ; p1 ! p2 ! p‘ !
where p1 ; p2 ; . . . ; p‘ are respectively the multiplicities of distinct coordinates ci1 > ci2 > > ci‘ of c 2 Rn . We introduce a real ternary form
F A ðt; x; yÞ ¼ detðtIn þ xRðAÞ þ yIðAÞÞ; where RðAÞ ¼ ðA þ A Þ=2 and IðAÞ ¼ ðA A Þ=ð2iÞ. The ternary form satisfies the equation
F A ðt; x; yÞ ¼
n Y
ðt kj ðxRðAÞ þ yIðAÞÞÞ
j¼1
⇑ Corresponding author. 1 2
E-mail addresses: [email protected] (M.T. Chien), [email protected] (H. Nakazato). Partially supported by Taiwan National Science Council under NSC 102-2115-M-031-001. Supported in part by Japan Society for Promotion of Science, KAKENHI (23540180).
http://dx.doi.org/10.1016/j.amc.2014.01.067 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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M.T. Chien, H. Nakazato / Applied Mathematics and Computation 232 (2014) 178–182
for every ðx; yÞ 2 R2 , where k1 ðÞ P k2 ðÞ P P kn ðÞ denote the decreasing eigenvalues of an n-by-n Hermitian matrix. It is well known that WðAÞ is the convex hull of the real affine part of the dual curve of F A ðt; x; yÞ ¼ 0. For the study of c-numerical range, a real ternary form GA;c ðt; x; yÞ of degree n!=ðp1 ! p2 ! . . . p‘ !Þ is defined in [4] by
GA;c ðt; x; yÞ ¼
Y
t
! X crðjÞ kj ðxRðAÞ þ yIðAÞÞ
r
rðjÞ
2
for every ðx; yÞ 2 R , and where r runs over the partitions of f1; 2; . . . ; ng with p1 ; p2 ; . . . ; p‘ elements. The form GA;c ðt; x; yÞ itself and its irreducible factors in C½t; x; y are hyperbolic with respect to ð1; 0; 0Þ, that is, the roots of the equation GA;c ðt; x; yÞ ¼ 0 in t are all real for every ðx; yÞ 2 R. For simplicity, we assume that GA;c ðt; x; yÞ is irreducible. If GA;c ðt; x; yÞ ¼ 0 is a rational curve, the paper [4] applies Henrion’s method [11] to construct a matrix B so that W c ðAÞ ¼ WðBÞ. This method consists of two steps. The first step, using Bezout resultant, constructs triple Hermitian matrices ðC 0 ; C 1 ; C 2 Þ satisfying
GA;c ðt; x; yÞ ¼ det ðC 0 Þ1 detðtC 0 þ xC 1 þ yC 2 Þ 1=2
1=2
with a positive definite Hermitian matrix C 0 . The second step is the construction of B as C 0 ðC 1 þ iC 2 ÞC 0 , which involves the square root of a positive definite matrix. In this paper, we use compound matrix method which is applicable to reduce the joint c-numerical ranges of m-tuple of Hermitian matrices. We recall a generalization of the c-numerical range, namely, joint c-numerical range. Let Hn be the real vector space of nby-n complex Hermitian matrices. For any c ¼ ðc1 ; c2 ; . . . ; cn Þ 2 Rn and an ordered m-tuple H ¼ ðH1 ; H2 ; . . . ; Hm Þ 2 Hm n , the joint c-numerical range of H is defined as the set
W c ðHÞ ¼ W c ðH1 ; H2 ; . . . ; Hm Þ ( ! ) n n n X X X ¼ cj hH1 nj ; nj i; cj hH2 nj ; nj i; . . . ; cj hHm nj ; nj i 2 Rm : fn1 ; n2 ; . . . ; nn g is an orthonormal basis for Cn : j¼1
j¼1
j¼1
If c ¼ ð1; . . . ; 1; 0; . . . ; 0Þ, with 1’s appearing k times, W c ðH1 ; H2 ; . . . ; Hm Þ reduces to the joint k-numerical range W k ðHÞ ¼ W k ðH1 ; H2 ; . . . ; Hm Þ. Moreover, for k ¼ 1, the joint k-numerical range is just the joint regular numerical WðH1 ; H2 ; . . . ; Hm Þ. The object of joint numerical range is useful to the study of robust control problems (cf. [5,9]), and the joint k-numerical range is closely connected to the quadratic system (cf. [16]). Sufficient conditions for the joint numerical range to be convex are also discussed in [9]. If m ¼ 2, the joint c-numerical range W c ðH1 ; H2 Þ is identified with the c-numerical range W c ðH1 þ iH2 Þ which is convex. However, the joint c-numerical range W c ðH1 ; H2 ; . . . ; Hm Þ is in general not convex. Even for the joint numerical range W ðH1 ; H2 ; . . . ; Hm Þ, there are examples of non-convexity for m ¼ 3; n ¼ 2 and m ¼ 4; n ¼ 3 (cf. [9]). Nevertheless, Au-Yeung and Tsing [1] proved the convexity of the range W c ðH1 ; H2 ; H3 Þ for n P 3. In this paper, we introduce compound matrix method to investigate the reduction problem of the joint c-numerical ranges of m-tuple of Hermitian matrices. In particular, the joint c-numerical range of triple Hermitian matrices is attainable by a c-numerical range of some triple Hermitian matrices. 2. Reduction Let A 2 M n , and c ¼ ðc1 ; c2 ; . . . ; cn Þ 2 Rn . The method used in [4] to prove W c ðAÞ ¼ WðBÞ for some matrix B is based on an affirmative solution of the Lax–Fiedler conjecture [10]. We introduce compound matrices as a tool to investigate the reduction problem of the joint c-numerical range. The kth compound matrix C k ðAÞ of A is an N-by-N matrix whose a; b entry is det A½ajb, where N ¼ n C k ¼ n!=ðk!ðn kÞ!Þ, a; b 2 Q k;n and
Q k;n ¼ fa ¼ ða1 ; . . . ; ak Þ : 1 6 a1 < < ak 6 ng with usual lexicographical order. The kth additive compound matrix Dk ðAÞ of A is defined as
Dk ðAÞ ¼
d C k ðI þ tAÞjt¼0 ; dt
equivalently, Dk ðAÞ is the linear term in
C k ðI þ tAÞ ¼ I þ tDk ðAÞ þ t 2 R þ : (For references on the compound matrices, see, for instance, [15, pp. 502–507], also [12,17].) Let fe1 ; e2 ; . . . ; en g be the standard orthonormal basis of Cn . We consider the tensor product Cnk of k copies of the Hilbert space Cn . For k vectors n1 ; n2 ; . . . ; nk in Cn , their exterior product n1 ^ n2 ^ nk 2 Cnk is defined as
1 X pffiffiffiffi sgnr nrð1Þ nrð2Þ nrðkÞ : k! r2Sk
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Let V k ðCÞ be the n C k -dimensional vector space spanned by exterior products
ei1 ^ ei2 ^ ^ eik ; 1 6 i1 < i2 < < ik 6 n. An element in V k ðCÞ is called a decomposable k-vector if it is expressed as the exterior product n1 ^ n2 ^ nk of some k vectors n1 ; n2 ; . . . ; nk in Cn . In the case n ¼ 2, it is easy to find the relation that
W ðc1 ;c2 Þ ðH1 ; H2 ; . . . ; Hm Þ ¼ ðc1 c2 ÞWðH1 ; H2 ; . . . ; Hm Þ þ c2 ðtrðH1 Þ; trðH2 Þ; . . . ; trðHm ÞÞ ¼ Wððc1 c2 ÞH1 þ c2 trðH1 ÞI2 ; ðc1 c2 ÞH2 þ c2 trðH2 ÞI2 . . . ; ðc1 c2 ÞHm þ c2 trðHm ÞI2 Þ: Therefore the joint c-numerical range of m-tuple of 2-by-2 Hermitian matrices is attainable by the c- numerical range of some m-tuple of 2-by-2 Hermitian matrices. Hence in the following of this paper, we always assume that the size of Hermitian matrices n P 3. n Theorem 1. Let H ¼ ðH1 ; H2 ; . . . ; Hm Þ 2 Hm in decreasing order with ‘ distinct coordinates of n , and c ¼ ðc 1 ; c 2 ; . . . ; cn Þ 2 R multiplicities p1 ; p2 ; . . . ; p‘ respectively. Then there exists m-tuple of N-by-N Hermitian matrices ðK 1 ; K 2 ; . . . ; K m Þ such that
W c ðHÞ WðK 1 ; K 2 ; . . . ; K m Þ;
ð1Þ
convðW c ðHÞÞ ¼ convðWðK 1 ; K 2 ; . . . ; K m ÞÞ;
ð2Þ
and
where
N ¼ n C p1 n C p1 þp2 n C p1 þþp‘1 : Proof. It is obvious that
W ðc1 þc0 ;c2 þc0 ;...;cn þc0 Þ ðH1 ; H2 ; . . . ; Hm Þ ¼ W ðc1 ;c2 ;...;cn Þ ðH1 ; H2 ; . . . ; Hm Þ þ c0 ðtrðH1 Þ; trðH2 Þ; . . . ; trðHm ÞÞ: Hence we may assume without loss of generality that cn ¼ 0. Define a sequence bi ¼ ci ciþ1 ; i ¼ 1; 2; . . . ; n 1. Then for any orthonormal basis fn1 ; n2 ; . . . ; nn g of Cn and j ¼ 1; 2; . . . ; m, we have that
c1 hHj n1 ; n1 i þ c2 hHj n2 ; n2 i þ þ cn1 hHj nn1 ; nn1 i þ cn hHj nn ; nn i ¼ ¼
X
n1 X bs ðhHj n1 ; n1 i þ hHj n2 ; n2 i þ þ hHj ns ; ns iÞ s¼1
bs ðhHj n1 ; n1 i þ hHj n2 ; n2 i þ þ hHj ns ; ns iÞ:
ð3Þ
s¼p1 ;p1 þp2 ;...;p1 þþp‘1
Eq. (3) provides a motivation to define the following Hermitian matrices, for j ¼ 1; 2; . . . ; m,
K j ¼ bp1 ðDp1 ðHj Þ Im2 Im3 Im‘1 Þ þ bp1 þp2 ðIm1 Dp1 þp2 ðHj Þ Im3 Im‘1 Þ þ bp1 þp2 þp3 ðIm1 Im2 Dp1 þp2 þp3 ðHj Þ Im‘1 Þ þ þ bp1 þþp‘1 ðIp1 Im1 Im‘2 Dp1 þþp‘1 ðHj ÞÞ;
ð4Þ
where ms ¼ n C p1 þp2 þþps ; s ¼ 1; 2; . . . ; ‘ 1. For each component Hermitian matrix Hj , the ði; jÞ-entry of Dk ðHj Þ is characterized as
Dk ðHj Þði1 ; i2 ; . . . ; ik : j1 ; j2 ; . . . ; jk Þ ¼ hDk ðHj Þej1 ^ ^ ejk ; ei1 ^ ^ eik i ¼
k X
hHj ejs ; eis i:
s¼1
This relation also holds for an arbitrary orthonormal basis fn1 ; n2 ; . . . ; nn g of Cn that
hDk ðHj Þnj1 ^ ^ njk ; n1 ^ ^ nk i ¼
k X hHj njs ; ns i:
ð5Þ
s¼1
By (5), Eq. (3) becomes
c1 hHj n1 ; n1 i þ c2 hHj n2 ; n2 i þ þ cn1 hHj nn1 ; nn1 i þ cn hHj nn ; nn i X ¼ bs hDs ðHj Þnj1 ^ ^ njs ; n1 ^ ^ ns i s¼p1 ;p1 þp2 ;...;p1 þþp‘1
¼ hK j ðn1 ^ ^ np1 n1 ^ ^ np1 þþp‘1 Þ; ðn1 ^ ^ np1 n1 ^ ^ np1 þþp‘1 Þi: Since the vector n1 ^ ^ np1 n1 ^ ^ np1 þþp‘1 is unital, the inclusion (1) then follows. A fundamental property proved in [15, Theorem F.5], shows that the eigenvalues of the kth additive compound of A are sums of k eigenvalues of A (see also [2]). Hence we have that
181
M.T. Chien, H. Nakazato / Applied Mathematics and Computation 232 (2014) 178–182 k X kj ðRðeih AÞÞ ¼ k1 ðRðeih Dk ðAÞÞÞ
ð6Þ
j¼1
for 0 6 h < 2p. We easily obtain a generalization of (6) that k X kj ðx1 H1 þ x2 H2 þ þ xm Hm Þ ¼ k1 ðDk ðx1 H1 þ x2 H2 þ þ xm Hm ÞÞ
ð7Þ
j¼1
for ðx1 ; x2 ; . . . ; xm Þ 2 Rm . By a result on the spectrum of the Kronecker sum of matrices in [13, Theorem 4.4.5], we obtain that
(
rðx1 K 1 þ x2 K 2 þ þ xm K m Þ ¼ t1 þ t2 þ t3 þ þ t‘1 : tj 2 ðbp1 þþpj Þr
m X
!
)
xq Dp1 þ...þpj ðHq Þ ;
j ¼ 1; 2; . . . ; ‘ 1 :
q¼1
Then, by equality (7), we have that ‘1 m X X k1 ðx1 K 1 þ x2 K 2 þ þ xm K m Þ ¼ bp1 þþpj k1 xq Dp1 þþpj ðHq Þ j¼1
q¼1
‘1 m X X bp1 þþpj k1 xq H q ¼ j¼1
!
! þ þ kp1 þþpj
q¼1
m X
!! xq H q
q¼1
n X cj kj ðx1 H1 þ x2 H2 þ þ xm Hm Þ: ¼
ð8Þ
j¼1
Suppose that /ðxÞ is the support function of convðW c ðHÞÞ. Then, for x ¼ ðx1 ; x2 ; . . . ; xm Þ 2 Rm , we have
/ðxÞ ¼ maxfhx; yi : y 2 convðW c ðHÞÞg ¼ maxfhx; yi : y 2 W c ðHÞg ( * + ) n m X X ¼ max cj xk Hk nj ; nj : fn1 ; n2 ; . . . ; nn g is an orthonormal basis for Cn j¼1
k¼1
n X cj kj ðx1 H1 þ x2 H2 þ þ xm Hm Þ ¼ k1 ðx1 K 1 þ x2 K 2 þ þ xm K m Þ; ¼
ðby ð8ÞÞ
j¼1
which is the support function of convðWðK 1 ; K 2 ; . . . ; K m ÞÞ. This proves that both convex sets convðW c ðHÞÞ and convðWðK 1 ; K 2 ; . . . ; K m ÞÞ have the same support function, and thus (2) follows. h The inclusion (1) is important for the joint c-numerical range W c ðHÞ and the joint regular numerical range WðK 1 ; K 2 ; . . . ; K m Þ, both ranges may not convex. We give an example to show that two sets may have the same convex hulls but do not contain each other. Consider two sets A ¼ fðx; yÞ : 0 6 x 6 1; 0 6 y 6 xg [ ð0; 1Þ and B ¼ fðx; yÞ : 0 6 x 6 1; 0 6 y 6 1 xg [ ð1; 1Þ. Then convðAÞ ¼ convðBÞ, but neither A # B nor B # A. As a consequence of Theorem 1, we have the following reduction of joint c-numerical ranges. n Theorem 2. Let H ¼ ðH1 ; H2 ; . . . ; Hm Þ 2 Hm in decreasing order with ‘ distinct coordinates of n , and c ¼ ðc 1 ; c2 ; . . . ; cn Þ 2 R multiplicities p1 ; p2 ; . . . ; p‘ respectively, and let K 1 ; K 2 ; . . . ; K m be defined as in (4). If W c ðHÞ is convex then W c ðHÞ ¼ WðK 1 ; K 2 ; . . . ; K m Þ.
Proof. If W c ðHÞ is convex then, by Theorem 1, we have that
W c ðHÞ WðK 1 ; K 2 ; . . . ; K m Þ convðWðK 1 ; K 2 ; . . . ; K m ÞÞ ¼ convðW c ðHÞÞ ¼ W c ðHÞ: In particular, for m ¼ 3, the following result is immediate. Theorem 3. Let H ¼ ðH1 ; H2 ; H3 Þ be a triple n-by-n Hermitian matrices, and c ¼ ðc1 ; c2 ; . . . ; cn Þ 2 Rn in decreasing order with ‘ distinct coordinates of multiplicities p1 ; p2 ; . . . ; p‘ respectively, and let K 1 ; K 2 ; K 3 be defined as in (4). Then W c ðH1 ; H2 ; H3 Þ ¼ WðK 1 ; K 2 ; K 3 Þ, and W k ðH1 ; H2 ; H3 Þ ¼ WðDk ðH1 Þ; Dk ðH2 Þ; Dk ðH3 ÞÞ for 1 6 k 6 n. Proof. It is shown in [1] that W c ðH1 ; H2 ; H3 Þ is convex, and thus by Theorem 2, we conclude that W c ðH1 ; H2 ; H3 Þ ¼ WðK 1 ; K 2 ; K 3 Þ. Moreover, for c ¼ ð1; . . . ; 1; 0; . . . ; 0Þ with 1’s appearing k times, then for j ¼ 1; 2; 3, the K j Hermitian matrices defined in (4) become
K j ¼ Dk ðHj Þ Im2 Im3 Im‘1 :
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M.T. Chien, H. Nakazato / Applied Mathematics and Computation 232 (2014) 178–182
Let A 2 M n and c 2 Rn with distinct coordinates c1 > c2 > > cn . It is proved in [4] that there exists an n!-by-n! matrix B satisfying W c ðAÞ ¼ WðBÞ. For n ¼ 3, a 6-by-6 matrix B exists so that W c ðAÞ ¼ WðBÞ. In the case of triple 3-by-3 Hermitian matrices H1 ; H2 ; H3 , Theorem 3 assures that there exist 9-by-9 Hermitian matrices K 1 ; K 2 ; K 3 such that W c ðH1 ; H2 ; H3 Þ ¼ WðK 1 ; K 2 ; K 3 Þ. For example, we consider a 3-by-3 Hermitian matrix
0
k1
B H ¼ @ a12
a13
1
a12 a13 C k2 a23 A: a23 k3
The additive compound matrix D2 ðHÞ is given by
0
a23
k1 þ k2
B D2 ðHÞ ¼ @ a23
a12 C A:
k1 þ k3
a13
1
a13
a12
k2 þ k3
Suppose that ðc1 ; c2 ; c3 Þ ¼ ðb þ 1; b; 0Þ with b ¼ 3, and
0
1 1 0 0 B C H 1 ¼ @ 0 0 0 A; 0 0 1
0
1 0 1 0 B C H2 ¼ @ 1 0 2 A; 0 2 0
0
1 0 1þi 1i B C H3 ¼ @ 1 i 0 1 i A: 1þi 1þi 0
Then, the construction K j ; j ¼ 1; 2; 3, in Theorem 3 becomes
K j ¼ Hj I3 þ 3I3 D2 ðHj Þ:
ð9Þ
For instance
K 1 ¼ diagð4; 1; 2; 3; 0; 3; 2; 1; 4Þ; 0
0 B6 B B B0 B B B1 B K2 ¼ B B0 B B0 B B0 B B @0
6 0 0
1 0
3 0
0
0
0
1 0
0
0
0
0
0C C C 0C C C 0C C 0C C; C 2C C 0C C C 3A 0
3 0
0
0 1
0
0
6 0 2 0
0
1 0
1
0
6 0 3 0 2
0
1 0
0 0
0 2 0 0 0 6 0 0 2 0 6 0
3 0
0
0
0 0 0 0 0 2 0 3
and K 3 can be found in the same way according to (9). It is natural to ask whether there exist a triple Hermitian matrices ðK 1 ; K 2 ; K 3 Þ of size less than 9, e.g. size 6, satisfying W c ðH1 ; H2 ; H3 Þ ¼ WðK 1 ; K 2 ; K 3 Þ. This problem remains open for further study. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Reduction of joint c-numerical ranges M.T. Chien a,⇑,1, H. Nakazato a,b,2 a b
Department of Mathematics, Soochow University, Taipei 11102, Taiwan Department of Mathematical Sciences, Faculty of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan
a r t i c l e
i n f o
a b s t r a c t Let ðH1 ; H2 ; . . . ; Hm Þ be an m-tuple of n-by-n Hermitian matrices and c 2 Rn . We investigate the reduction problem of the joint c-numerical range W c ðH1 ; H2 ; . . . ; Hm Þ. In particular, for m ¼ 3 and n P 3, we prove that W c ðH1 ; H2 ; H3 Þ is attainable by the joint numerical range WðK 1 ; K 2 ; K 3 Þ for some triple Hermitian matrices ðK 1 ; K 2 ; K 3 Þ. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Joint c-numerical range Reduction Compound matrix Robust control
1. Introduction Let A 2 M n , the algebra of n-by-n complex matrices. For a real n-tuple c ¼ ðc1 ; c2 ; . . . ; cn Þ, the c-numerical range W c ðAÞ of A is defined as the set
( ) n X n W c ðAÞ ¼ cj xj Axj : fx1 ; x2 ; . . . ; xn g is an orthonormal basis for C : j¼1
Westwick [18] proved the convexity of the range W c ðAÞ. If c ¼ ð1; . . . ; 1; 0; . . . ; 0Þ, with 1’s appearing k times, W c ðAÞ becomes the k-numerical range W k ðAÞ. In particular, k ¼ 1; W k ðAÞ reduces to the classical numerical range which is usually denoted by WðAÞ. The subject of various numerical ranges has been extensively developed and studied (see, for instance, [8,13]), and has connections to many different branches of science, such as quantum mechanics [7] and stability theory [6,14]. It is shown in [3] that for some nilpotent Toeplitz matrix A 2 M n and any c 2 Rn , there exists an n!-by-n! matrix B satisfying W c ðAÞ ¼ WðBÞ. Furthermore, the paper [4] proves that this reduction property is also true for arbitrary matrix A 2 M n , and the size of B can be decreased to
n
p1 p2 p‘
¼
n! ; p1 ! p2 ! p‘ !
where p1 ; p2 ; . . . ; p‘ are respectively the multiplicities of distinct coordinates ci1 > ci2 > > ci‘ of c 2 Rn . We introduce a real ternary form
F A ðt; x; yÞ ¼ detðtIn þ xRðAÞ þ yIðAÞÞ; where RðAÞ ¼ ðA þ A Þ=2 and IðAÞ ¼ ðA A Þ=ð2iÞ. The ternary form satisfies the equation
F A ðt; x; yÞ ¼
n Y
ðt kj ðxRðAÞ þ yIðAÞÞÞ
j¼1
⇑ Corresponding author. 1 2
E-mail addresses: [email protected] (M.T. Chien), [email protected] (H. Nakazato). Partially supported by Taiwan National Science Council under NSC 102-2115-M-031-001. Supported in part by Japan Society for Promotion of Science, KAKENHI (23540180).
http://dx.doi.org/10.1016/j.amc.2014.01.067 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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for every ðx; yÞ 2 R2 , where k1 ðÞ P k2 ðÞ P P kn ðÞ denote the decreasing eigenvalues of an n-by-n Hermitian matrix. It is well known that WðAÞ is the convex hull of the real affine part of the dual curve of F A ðt; x; yÞ ¼ 0. For the study of c-numerical range, a real ternary form GA;c ðt; x; yÞ of degree n!=ðp1 ! p2 ! . . . p‘ !Þ is defined in [4] by
GA;c ðt; x; yÞ ¼
Y
t
! X crðjÞ kj ðxRðAÞ þ yIðAÞÞ
r
rðjÞ
2
for every ðx; yÞ 2 R , and where r runs over the partitions of f1; 2; . . . ; ng with p1 ; p2 ; . . . ; p‘ elements. The form GA;c ðt; x; yÞ itself and its irreducible factors in C½t; x; y are hyperbolic with respect to ð1; 0; 0Þ, that is, the roots of the equation GA;c ðt; x; yÞ ¼ 0 in t are all real for every ðx; yÞ 2 R. For simplicity, we assume that GA;c ðt; x; yÞ is irreducible. If GA;c ðt; x; yÞ ¼ 0 is a rational curve, the paper [4] applies Henrion’s method [11] to construct a matrix B so that W c ðAÞ ¼ WðBÞ. This method consists of two steps. The first step, using Bezout resultant, constructs triple Hermitian matrices ðC 0 ; C 1 ; C 2 Þ satisfying
GA;c ðt; x; yÞ ¼ det ðC 0 Þ1 detðtC 0 þ xC 1 þ yC 2 Þ 1=2
1=2
with a positive definite Hermitian matrix C 0 . The second step is the construction of B as C 0 ðC 1 þ iC 2 ÞC 0 , which involves the square root of a positive definite matrix. In this paper, we use compound matrix method which is applicable to reduce the joint c-numerical ranges of m-tuple of Hermitian matrices. We recall a generalization of the c-numerical range, namely, joint c-numerical range. Let Hn be the real vector space of nby-n complex Hermitian matrices. For any c ¼ ðc1 ; c2 ; . . . ; cn Þ 2 Rn and an ordered m-tuple H ¼ ðH1 ; H2 ; . . . ; Hm Þ 2 Hm n , the joint c-numerical range of H is defined as the set
W c ðHÞ ¼ W c ðH1 ; H2 ; . . . ; Hm Þ ( ! ) n n n X X X ¼ cj hH1 nj ; nj i; cj hH2 nj ; nj i; . . . ; cj hHm nj ; nj i 2 Rm : fn1 ; n2 ; . . . ; nn g is an orthonormal basis for Cn : j¼1
j¼1
j¼1
If c ¼ ð1; . . . ; 1; 0; . . . ; 0Þ, with 1’s appearing k times, W c ðH1 ; H2 ; . . . ; Hm Þ reduces to the joint k-numerical range W k ðHÞ ¼ W k ðH1 ; H2 ; . . . ; Hm Þ. Moreover, for k ¼ 1, the joint k-numerical range is just the joint regular numerical WðH1 ; H2 ; . . . ; Hm Þ. The object of joint numerical range is useful to the study of robust control problems (cf. [5,9]), and the joint k-numerical range is closely connected to the quadratic system (cf. [16]). Sufficient conditions for the joint numerical range to be convex are also discussed in [9]. If m ¼ 2, the joint c-numerical range W c ðH1 ; H2 Þ is identified with the c-numerical range W c ðH1 þ iH2 Þ which is convex. However, the joint c-numerical range W c ðH1 ; H2 ; . . . ; Hm Þ is in general not convex. Even for the joint numerical range W ðH1 ; H2 ; . . . ; Hm Þ, there are examples of non-convexity for m ¼ 3; n ¼ 2 and m ¼ 4; n ¼ 3 (cf. [9]). Nevertheless, Au-Yeung and Tsing [1] proved the convexity of the range W c ðH1 ; H2 ; H3 Þ for n P 3. In this paper, we introduce compound matrix method to investigate the reduction problem of the joint c-numerical ranges of m-tuple of Hermitian matrices. In particular, the joint c-numerical range of triple Hermitian matrices is attainable by a c-numerical range of some triple Hermitian matrices. 2. Reduction Let A 2 M n , and c ¼ ðc1 ; c2 ; . . . ; cn Þ 2 Rn . The method used in [4] to prove W c ðAÞ ¼ WðBÞ for some matrix B is based on an affirmative solution of the Lax–Fiedler conjecture [10]. We introduce compound matrices as a tool to investigate the reduction problem of the joint c-numerical range. The kth compound matrix C k ðAÞ of A is an N-by-N matrix whose a; b entry is det A½ajb, where N ¼ n C k ¼ n!=ðk!ðn kÞ!Þ, a; b 2 Q k;n and
Q k;n ¼ fa ¼ ða1 ; . . . ; ak Þ : 1 6 a1 < < ak 6 ng with usual lexicographical order. The kth additive compound matrix Dk ðAÞ of A is defined as
Dk ðAÞ ¼
d C k ðI þ tAÞjt¼0 ; dt
equivalently, Dk ðAÞ is the linear term in
C k ðI þ tAÞ ¼ I þ tDk ðAÞ þ t 2 R þ : (For references on the compound matrices, see, for instance, [15, pp. 502–507], also [12,17].) Let fe1 ; e2 ; . . . ; en g be the standard orthonormal basis of Cn . We consider the tensor product Cnk of k copies of the Hilbert space Cn . For k vectors n1 ; n2 ; . . . ; nk in Cn , their exterior product n1 ^ n2 ^ nk 2 Cnk is defined as
1 X pffiffiffiffi sgnr nrð1Þ nrð2Þ nrðkÞ : k! r2Sk
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Let V k ðCÞ be the n C k -dimensional vector space spanned by exterior products
ei1 ^ ei2 ^ ^ eik ; 1 6 i1 < i2 < < ik 6 n. An element in V k ðCÞ is called a decomposable k-vector if it is expressed as the exterior product n1 ^ n2 ^ nk of some k vectors n1 ; n2 ; . . . ; nk in Cn . In the case n ¼ 2, it is easy to find the relation that
W ðc1 ;c2 Þ ðH1 ; H2 ; . . . ; Hm Þ ¼ ðc1 c2 ÞWðH1 ; H2 ; . . . ; Hm Þ þ c2 ðtrðH1 Þ; trðH2 Þ; . . . ; trðHm ÞÞ ¼ Wððc1 c2 ÞH1 þ c2 trðH1 ÞI2 ; ðc1 c2 ÞH2 þ c2 trðH2 ÞI2 . . . ; ðc1 c2 ÞHm þ c2 trðHm ÞI2 Þ: Therefore the joint c-numerical range of m-tuple of 2-by-2 Hermitian matrices is attainable by the c- numerical range of some m-tuple of 2-by-2 Hermitian matrices. Hence in the following of this paper, we always assume that the size of Hermitian matrices n P 3. n Theorem 1. Let H ¼ ðH1 ; H2 ; . . . ; Hm Þ 2 Hm in decreasing order with ‘ distinct coordinates of n , and c ¼ ðc 1 ; c 2 ; . . . ; cn Þ 2 R multiplicities p1 ; p2 ; . . . ; p‘ respectively. Then there exists m-tuple of N-by-N Hermitian matrices ðK 1 ; K 2 ; . . . ; K m Þ such that
W c ðHÞ WðK 1 ; K 2 ; . . . ; K m Þ;
ð1Þ
convðW c ðHÞÞ ¼ convðWðK 1 ; K 2 ; . . . ; K m ÞÞ;
ð2Þ
and
where
N ¼ n C p1 n C p1 þp2 n C p1 þþp‘1 : Proof. It is obvious that
W ðc1 þc0 ;c2 þc0 ;...;cn þc0 Þ ðH1 ; H2 ; . . . ; Hm Þ ¼ W ðc1 ;c2 ;...;cn Þ ðH1 ; H2 ; . . . ; Hm Þ þ c0 ðtrðH1 Þ; trðH2 Þ; . . . ; trðHm ÞÞ: Hence we may assume without loss of generality that cn ¼ 0. Define a sequence bi ¼ ci ciþ1 ; i ¼ 1; 2; . . . ; n 1. Then for any orthonormal basis fn1 ; n2 ; . . . ; nn g of Cn and j ¼ 1; 2; . . . ; m, we have that
c1 hHj n1 ; n1 i þ c2 hHj n2 ; n2 i þ þ cn1 hHj nn1 ; nn1 i þ cn hHj nn ; nn i ¼ ¼
X
n1 X bs ðhHj n1 ; n1 i þ hHj n2 ; n2 i þ þ hHj ns ; ns iÞ s¼1
bs ðhHj n1 ; n1 i þ hHj n2 ; n2 i þ þ hHj ns ; ns iÞ:
ð3Þ
s¼p1 ;p1 þp2 ;...;p1 þþp‘1
Eq. (3) provides a motivation to define the following Hermitian matrices, for j ¼ 1; 2; . . . ; m,
K j ¼ bp1 ðDp1 ðHj Þ Im2 Im3 Im‘1 Þ þ bp1 þp2 ðIm1 Dp1 þp2 ðHj Þ Im3 Im‘1 Þ þ bp1 þp2 þp3 ðIm1 Im2 Dp1 þp2 þp3 ðHj Þ Im‘1 Þ þ þ bp1 þþp‘1 ðIp1 Im1 Im‘2 Dp1 þþp‘1 ðHj ÞÞ;
ð4Þ
where ms ¼ n C p1 þp2 þþps ; s ¼ 1; 2; . . . ; ‘ 1. For each component Hermitian matrix Hj , the ði; jÞ-entry of Dk ðHj Þ is characterized as
Dk ðHj Þði1 ; i2 ; . . . ; ik : j1 ; j2 ; . . . ; jk Þ ¼ hDk ðHj Þej1 ^ ^ ejk ; ei1 ^ ^ eik i ¼
k X
hHj ejs ; eis i:
s¼1
This relation also holds for an arbitrary orthonormal basis fn1 ; n2 ; . . . ; nn g of Cn that
hDk ðHj Þnj1 ^ ^ njk ; n1 ^ ^ nk i ¼
k X hHj njs ; ns i:
ð5Þ
s¼1
By (5), Eq. (3) becomes
c1 hHj n1 ; n1 i þ c2 hHj n2 ; n2 i þ þ cn1 hHj nn1 ; nn1 i þ cn hHj nn ; nn i X ¼ bs hDs ðHj Þnj1 ^ ^ njs ; n1 ^ ^ ns i s¼p1 ;p1 þp2 ;...;p1 þþp‘1
¼ hK j ðn1 ^ ^ np1 n1 ^ ^ np1 þþp‘1 Þ; ðn1 ^ ^ np1 n1 ^ ^ np1 þþp‘1 Þi: Since the vector n1 ^ ^ np1 n1 ^ ^ np1 þþp‘1 is unital, the inclusion (1) then follows. A fundamental property proved in [15, Theorem F.5], shows that the eigenvalues of the kth additive compound of A are sums of k eigenvalues of A (see also [2]). Hence we have that
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M.T. Chien, H. Nakazato / Applied Mathematics and Computation 232 (2014) 178–182 k X kj ðRðeih AÞÞ ¼ k1 ðRðeih Dk ðAÞÞÞ
ð6Þ
j¼1
for 0 6 h < 2p. We easily obtain a generalization of (6) that k X kj ðx1 H1 þ x2 H2 þ þ xm Hm Þ ¼ k1 ðDk ðx1 H1 þ x2 H2 þ þ xm Hm ÞÞ
ð7Þ
j¼1
for ðx1 ; x2 ; . . . ; xm Þ 2 Rm . By a result on the spectrum of the Kronecker sum of matrices in [13, Theorem 4.4.5], we obtain that
(
rðx1 K 1 þ x2 K 2 þ þ xm K m Þ ¼ t1 þ t2 þ t3 þ þ t‘1 : tj 2 ðbp1 þþpj Þr
m X
!
)
xq Dp1 þ...þpj ðHq Þ ;
j ¼ 1; 2; . . . ; ‘ 1 :
q¼1
Then, by equality (7), we have that ‘1 m X X k1 ðx1 K 1 þ x2 K 2 þ þ xm K m Þ ¼ bp1 þþpj k1 xq Dp1 þþpj ðHq Þ j¼1
q¼1
‘1 m X X bp1 þþpj k1 xq H q ¼ j¼1
!
! þ þ kp1 þþpj
q¼1
m X
!! xq H q
q¼1
n X cj kj ðx1 H1 þ x2 H2 þ þ xm Hm Þ: ¼
ð8Þ
j¼1
Suppose that /ðxÞ is the support function of convðW c ðHÞÞ. Then, for x ¼ ðx1 ; x2 ; . . . ; xm Þ 2 Rm , we have
/ðxÞ ¼ maxfhx; yi : y 2 convðW c ðHÞÞg ¼ maxfhx; yi : y 2 W c ðHÞg ( * + ) n m X X ¼ max cj xk Hk nj ; nj : fn1 ; n2 ; . . . ; nn g is an orthonormal basis for Cn j¼1
k¼1
n X cj kj ðx1 H1 þ x2 H2 þ þ xm Hm Þ ¼ k1 ðx1 K 1 þ x2 K 2 þ þ xm K m Þ; ¼
ðby ð8ÞÞ
j¼1
which is the support function of convðWðK 1 ; K 2 ; . . . ; K m ÞÞ. This proves that both convex sets convðW c ðHÞÞ and convðWðK 1 ; K 2 ; . . . ; K m ÞÞ have the same support function, and thus (2) follows. h The inclusion (1) is important for the joint c-numerical range W c ðHÞ and the joint regular numerical range WðK 1 ; K 2 ; . . . ; K m Þ, both ranges may not convex. We give an example to show that two sets may have the same convex hulls but do not contain each other. Consider two sets A ¼ fðx; yÞ : 0 6 x 6 1; 0 6 y 6 xg [ ð0; 1Þ and B ¼ fðx; yÞ : 0 6 x 6 1; 0 6 y 6 1 xg [ ð1; 1Þ. Then convðAÞ ¼ convðBÞ, but neither A # B nor B # A. As a consequence of Theorem 1, we have the following reduction of joint c-numerical ranges. n Theorem 2. Let H ¼ ðH1 ; H2 ; . . . ; Hm Þ 2 Hm in decreasing order with ‘ distinct coordinates of n , and c ¼ ðc 1 ; c2 ; . . . ; cn Þ 2 R multiplicities p1 ; p2 ; . . . ; p‘ respectively, and let K 1 ; K 2 ; . . . ; K m be defined as in (4). If W c ðHÞ is convex then W c ðHÞ ¼ WðK 1 ; K 2 ; . . . ; K m Þ.
Proof. If W c ðHÞ is convex then, by Theorem 1, we have that
W c ðHÞ WðK 1 ; K 2 ; . . . ; K m Þ convðWðK 1 ; K 2 ; . . . ; K m ÞÞ ¼ convðW c ðHÞÞ ¼ W c ðHÞ: In particular, for m ¼ 3, the following result is immediate. Theorem 3. Let H ¼ ðH1 ; H2 ; H3 Þ be a triple n-by-n Hermitian matrices, and c ¼ ðc1 ; c2 ; . . . ; cn Þ 2 Rn in decreasing order with ‘ distinct coordinates of multiplicities p1 ; p2 ; . . . ; p‘ respectively, and let K 1 ; K 2 ; K 3 be defined as in (4). Then W c ðH1 ; H2 ; H3 Þ ¼ WðK 1 ; K 2 ; K 3 Þ, and W k ðH1 ; H2 ; H3 Þ ¼ WðDk ðH1 Þ; Dk ðH2 Þ; Dk ðH3 ÞÞ for 1 6 k 6 n. Proof. It is shown in [1] that W c ðH1 ; H2 ; H3 Þ is convex, and thus by Theorem 2, we conclude that W c ðH1 ; H2 ; H3 Þ ¼ WðK 1 ; K 2 ; K 3 Þ. Moreover, for c ¼ ð1; . . . ; 1; 0; . . . ; 0Þ with 1’s appearing k times, then for j ¼ 1; 2; 3, the K j Hermitian matrices defined in (4) become
K j ¼ Dk ðHj Þ Im2 Im3 Im‘1 :
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Let A 2 M n and c 2 Rn with distinct coordinates c1 > c2 > > cn . It is proved in [4] that there exists an n!-by-n! matrix B satisfying W c ðAÞ ¼ WðBÞ. For n ¼ 3, a 6-by-6 matrix B exists so that W c ðAÞ ¼ WðBÞ. In the case of triple 3-by-3 Hermitian matrices H1 ; H2 ; H3 , Theorem 3 assures that there exist 9-by-9 Hermitian matrices K 1 ; K 2 ; K 3 such that W c ðH1 ; H2 ; H3 Þ ¼ WðK 1 ; K 2 ; K 3 Þ. For example, we consider a 3-by-3 Hermitian matrix
0
k1
B H ¼ @ a12
a13
1
a12 a13 C k2 a23 A: a23 k3
The additive compound matrix D2 ðHÞ is given by
0
a23
k1 þ k2
B D2 ðHÞ ¼ @ a23
a12 C A:
k1 þ k3
a13
1
a13
a12
k2 þ k3
Suppose that ðc1 ; c2 ; c3 Þ ¼ ðb þ 1; b; 0Þ with b ¼ 3, and
0
1 1 0 0 B C H 1 ¼ @ 0 0 0 A; 0 0 1
0
1 0 1 0 B C H2 ¼ @ 1 0 2 A; 0 2 0
0
1 0 1þi 1i B C H3 ¼ @ 1 i 0 1 i A: 1þi 1þi 0
Then, the construction K j ; j ¼ 1; 2; 3, in Theorem 3 becomes
K j ¼ Hj I3 þ 3I3 D2 ðHj Þ:
ð9Þ
For instance
K 1 ¼ diagð4; 1; 2; 3; 0; 3; 2; 1; 4Þ; 0
0 B6 B B B0 B B B1 B K2 ¼ B B0 B B0 B B0 B B @0
6 0 0
1 0
3 0
0
0
0
1 0
0
0
0
0
0C C C 0C C C 0C C 0C C; C 2C C 0C C C 3A 0
3 0
0
0 1
0
0
6 0 2 0
0
1 0
1
0
6 0 3 0 2
0
1 0
0 0
0 2 0 0 0 6 0 0 2 0 6 0
3 0
0
0
0 0 0 0 0 2 0 3
and K 3 can be found in the same way according to (9). It is natural to ask whether there exist a triple Hermitian matrices ðK 1 ; K 2 ; K 3 Þ of size less than 9, e.g. size 6, satisfying W c ðH1 ; H2 ; H3 Þ ¼ WðK 1 ; K 2 ; K 3 Þ. This problem remains open for further study. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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