We consider the extremal inertias and ranks of the matrix expressions f(X, Y) = A₃ − B₃X − (B₃X) ^* − C₃YD₃ − (C₃YD₃) ^*, where B₃, C₃, and D₃ are known matrices and Y and X are the solutions to the matrix equations A₁Y = C₁, YB₁ = D₁, and A₂X = C₂, respectively. As applications, we present necessary and sufficient condition for the previous matrix function f(X, Y) to be positive (negative), non-negative (positive) definite or nonsingular. We also characterize the relations between the Hermitian part of the solutions of the above-mentioned matrix equations. Furthermore, we establish necessary and sufficient conditions for the solvability of the system of matrix equations A₁Y = C₁, YB₁ = D₁, A₂X = C₂, and B₃X + (B₃X) ^* + C₃YD₃ + (C₃YD₃) ^* = A₃, and give an expression of the general solution to the above-mentioned system when it is solvable.

1. Introduction

Throughout, we denote the field of complex numbers by ℂ, the set of all m × n matrices over ℂ by ℂ^m×n, and the set of all m × m Hermitian matrices by

. The symbols A^* and ℛ(A) stand for the conjugate transpose, the column space of a complex matrix A respectively. I_n denotes the n × n identity matrix. The Moore-Penrose inverse [1] A^† of A, is the unique solution X to the four matrix equations:

(1)

Moreover, L_A and R_A stand for the projectors L_A = I − A^†A, R_A = I − AA^† induced by A. It is well known that the eigenvalues of a Hermitian matrix A ∈ ℂ^n×n are real, and the inertia of A is defined to be the triplet

(2)

where i₊(A), i₋(A), and i₀(A) stand for the numbers of positive, negative, and zero eigenvalues of A, respectively. The symbols i₊(A) and i₋(A) are called the positive index and the negative index of inertia, respectively. For two Hermitian matrices A and B of the same sizes, we say A ≥ B (A ≤ B) in the Löwner partial ordering if A − B is positive (negative) semidefinite. The Hermitian part of X is defined as H(X) = X + X^*. We will say that X is Re-nnd (Re-nonnegative semidefinite) if H(X) ≥ 0, X is Re-pd (Re-positive definite) if H(X) > 0, and X is Re-ns if H(X) is nonsingular.

It is well known that investigation on the solvability conditions and the general solution to linear matrix equations is very active (e.g., [2–9]). In 1999, Braden [10] gave the general solution to

(3)

In 2007, Djordjević [11] considered the explicit solution to (3) for linear bounded operators on Hilbert spaces. Moreover, Cao [12] investigated the general explicit solution to

(4)

Xu et al. [13] obtained the general expression of the solution of operator equation (4). In 2012, Wang and He [14] studied some necessary and sufficient conditions for the consistence of the matrix equation

(5)

and presented an expression of the general solution to (5).

Note that (5) is a special case of the following system:

(6)

To our knowledge, there has been little information about (6). One goal of this paper is to give some necessary and sufficient conditions for the solvability of the system of matrix (6) and present an expression of the general solution to system (6) when it is solvable.

In order to find necessary and sufficient conditions for the solvability of the system of matrix equations (6), we need to consider the extremal ranks and inertias of (10) subject to (13) and (11).

There have been many papers to discuss the extremal ranks and inertias of the following Hermitian expressions:

(7)

(8)

(9)

(10)

Tian has contributed much in this field. One of his works [15] considered the extremal ranks and inertias of (7). He and Wang [16] derived the extremal ranks and inertias of (7) subject to A₁X = C₁, A₂XB₂ = C₂. Liu and Tian [17] studied the extremal ranks and inertias of (8). Chu et al. [18] and Liu and Tian [19] derived the extremal ranks and inertias of (9). Zhang et al. [20] presented the extremal ranks and inertias of (9), where X and Y are Hermitian solutions of

(11)

(12)

respectively. He and Wang [16] derived the extremal ranks and inertias of (10). We consider the extremal ranks and inertias of (10) subject to (11) and

(13)

which is not only the generalization of the above matrix functions, but also can be used to investigate the solvability conditions for the existence of the general solution to the system (6). Moreover, it can be applied to characterize the relations between Hermitian part of the solutions of (11) and (13).

The remainder of this paper is organized as follows. In Section 2, we consider the extremal ranks and inertias of (10) subject to (11) and (13). In Section 3, we characterize the relations between the Hermitian part of the solution to (11) and (13). In Section 4, we establish the solvability conditions for the existence of a solution to (6) and obtain an expression of the general solution to (6).

2. Extremal Ranks and Inertias of Hermitian Matrix Function (10) with Some Restrictions

In this section, we consider formulas for the extremal ranks and inertias of (10) subject to (11) and (13). We begin with the following Lemmas.

Lemma 1 (see [21].)(a) Let A₁, C₁, B₁, and D₁ be given. Then the following statements are equivalent:

(1)
system (13) is consistent,
(2)

(14)
(3)

(15)

In this case, the general solution can be written as

(16)

where V is arbitrary.

(b) Let A₂ and C₂ be given. Then the following statements are equivalent:

(1)
equation (11) is consistent,
(2)

(17)
(3)

(18)

In this case, the general solution can be written as

(19)

where W is arbitrary.

Lemma 2 ([22, Lemma 1.5, Theorem 2.3]). Let , B ∈ ℂ^m×n, and , and denote that

(20)

Then one has the following

(a)
the following equalities hold
(21)

(22)
(b)
if ℛ(B)⊆ℛ(A), then i_±(L) = i_±(A) + i_±(D − B^*A^†B)). Thus i_±(L) = i_±(A) if and only if ℛ(B)⊆ℛ(A) and i_±(D − B^*A^†B) = 0,
(c)

(23)

Lemma 3 (see [23].)Let A ∈ ℂ^m×n, B ∈ ℂ^m×k, and C ∈ ℂ^l×n. Then they satisfy the following rank equalities:

(a)
r[A B] = r(A) + r(E_AB) = r(B) + r(E_BA),
(b)
,
(c)
,
(d)
,
(e)
,
(f)
,

Lemma 4 (see [15].)Let , B ∈ ℂ^m×n, , Q ∈ ℂ^m×n, and P ∈ ℂ^p×n be given, and T ∈ ℂ^m×m be nonsingular. Then one has the following

(1)
i_±(TAT^*) = i_±(A),
(2)
,
(3)
,
(4)
.

Lemma 5 (see [22], Lemma 1.4.)Let S be a set consisting of (square) matrices over ℂ^m×m, and let H be a set consisting of (square) matrices over . Then Then one has the following

(a)
S has a nonsingular matrix if and only if max _X∈S r(X) = m;
(b)
any X ∈ S is nonsingular if and only if min _X∈S r(X) = m;
(c)
{0} ∈ S if and only if min _X∈S r(X) = 0;
(d)
S = {0} if and only if max _X∈S r(X) = 0;
(e)
H has a matrix X > 0 (X < 0) if and only if max _X∈H i₊(X) = m (max _X∈H i₋(X) = m);
(f)
any X ∈ H satisfies X > 0 (X < 0) if and only if min _X∈H i₊(X) = m (min _X∈H i₋(X) = m);
(g)
H has a matrix X ≥ 0 (X ≤ 0) if and only if min _X∈H i₋(X) = 0 (min _X∈H i₊(X) = 0);
(h)
any X ∈ H satisfies X ≥ 0 (X ≤ 0) if and only if max _X∈H i₋(X) = 0 (max _X∈H i₊(X) = 0).

Lemma 6 (see [16].)Let p(X, Y) = A − BX − (BX) ^* − CYD − (CYD) ^*, where A, B, C, and D are given with appropriate sizes, and denote that

(24)

Then one has the following:

(1)
the maximal rank of p(X, Y) is
(25)
(2)
the minimal rank of p(X, Y) is
(26)
(3)
the maximal inertia of p(X, Y) is
(27)
(4)
the minimal inertias of p(X, Y) is
(28)

where

(29)

Now we present the main theorem of this section.

Theorem 7. Let A₁ ∈ ℂ^m×n, C₁ ∈ ℂ^m×k, B₁ ∈ ℂ^k×l, D₁ ∈ ℂ^n×l, A₂ ∈ ℂ^t×q, C₂ ∈ ℂ^t×p, , B₃ ∈ ℂ^p×q, C₃ ∈ ℂ^p×n, and D₃ ∈ ℂ^p×n be given, and suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by S and (11) by G. Put

(30)

Then one has the following:

(a)
the maximal rank of (10) subject to (13) and (11) is
(31)
(b)
the minimal rank of (10) subject to (13) and (11) is
(32)
(c)
the maximal inertia of (10) subject to (13) and (11) is
(33)
(d)
the minimal inertia of (10) subject to (13) and (11) is
(34)

Proof. By Lemma 1, the general solutions to (13) and (11) can be written as

(35)

where W and Z are arbitrary matrices with appropriate sizes. Put

(36)

Substituting (36) into (10) yields

(37)

Clearly P is Hermitian. It follows from Lemma 6 that

(38)

(39)

(40)

(41)

where

(42)

Now, we simplify the ranks and inertias of block matrices in (38)–(41).

By Lemma 4, block Gaussian elimination, and noting that

(43)

we have the following:

(44)

By C₁B₁ = A₁D₁, we obtain

(45)

By Lemma 2, we can get the following:

(46)

(47)

Substituting (44)-(47) into (38) and (41) yields (31)–(34), respectively.

Corollary 8. Let A₁, C₁, B₁, D₁, A₂, C₂, A₃, B₃, C₃, D₃, and E_i, (i = 1,2, …, 5) be as in Theorem 7, and suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by S and (11) by G. Then, one has the following:

(a)
there exist X ∈ G and Y ∈ S such that A₃ − B₃X − (B₃X) ^* − C₃YD₃ − (C₃YD₃) ^* > 0 if and only if
(48)
(b)
there exist X ∈ G and Y ∈ S such that A₃ − B₃X − (B₃X) ^* − C₃YD₃ − (C₃YD₃) ^* < 0 if and only if
(49)
(c)
there exist X ∈ G and Y ∈ S such that A₃ − B₃X − (B₃X) ^* − C₃YD₃ − (C₃YD₃) ^* ≥ 0 if and only if
(50)
(d)
there exist X ∈ G and Y ∈ S such that A₃ − B₃X − (B₃X) ^* − C₃YD₃ − (C₃YD₃) ^* ≤ 0 if and only if
(51)
(e)
A₃ − B₃X − (B₃X) ^* − C₃YD₃ − (C₃YD₃) ^* > 0 for all X ∈ G and Y ∈ S if and only if
(52)
(f)
A₃ − B₃X − (B₃X) ^* − C₃YD₃ − (C₃YD₃) ^* < 0 for all X ∈ G and Y ∈ S if and only if
(53)
(g)
A₃ − B₃X − (B₃X) ^* − C₃YD₃ − (C₃YD₃) ^* ≥ 0 for all X ∈ G and Y ∈ S if and only if
(54)
(h)
A₃ − B₃X − (B₃X) ^* − C₃YD₃ − (C₃YD₃) ^* ≤ 0 for all X ∈ G and Y ∈ S if and only if
(55)
(i)
there exist X ∈ G and Y ∈ S such that A₃ − B₃X − (B₃X) ^* − C₃YD₃ − (C₃YD₃) ^* is nonsingular if and only if
(56)

3. Relations between the Hermitian Part of the Solutions to (13) and (11)

Now we consider the extremal ranks and inertias of the difference between the Hermitian part of the solutions to (13) and (11).

Theorem 9. Let A₁ ∈ ℂ^m×p, C₁ ∈ ℂ^m×p, B₁ ∈ ℂ^p×l, D₁ ∈ ℂ^p×l, A₂ ∈ ℂ^t×p, and C₂ ∈ ℂ^t×p, be given. Suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by S and (11) by G. Put

(57)

Then one has the following:

(58)

Proof. By letting A₃ = 0, B₃ = −I, C₃ = I, and D₃ = I in Theorem 7, we can get the results.

Corollary 10. Let A₁, C₁, B₁, D₁, A₂, C₂, and H_i, (i = 1,2, …, 5) be as in Theorem 9, and suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by S and (11) by G. Then, one has the following:

(a)
there exist X ∈ G and Y ∈ S such that (X + X^*)>(Y + Y^*) if and only if
(59)
(b)
there exist X ∈ G and Y ∈ S such that (X + X^*)<(Y + Y^*) if and only if
(60)
(c)
there exist X ∈ G and Y ∈ S such that (X + X^*)≥(Y + Y^*) if and only if
(61)
(d)
there exist X ∈ G and Y ∈ S such that (X + X^*)≤(Y + Y^*) if and only if
(62)
(e)
(X + X^*)>(Y + Y^*) for all X ∈ G and Y ∈ S if and only if
(63)
(f)
(X + X^*)<(Y + Y^*) for all X ∈ G and Y ∈ S if and only if
(64)
(g)
(X + X^*)≥(Y + Y^*) for all X ∈ G and Y ∈ S if and only if
(65)
(h)
(X + X^*)≤(Y + Y^*) for all X ∈ G and Y ∈ S if and only if
(66)
(i)
there exist X ∈ G and Y ∈ S such that (X + X^*)−(Y + Y^*) is nonsingular if and only if
(67)

4. The Solvability Conditions and the General Solution to System (6)

We now turn our attention to (6). We in this section use Theorem 9 to give some necessary and sufficient conditions for the existence of a solution to (6) and present an expression of the general solution to (6). We begin with a lemma which is used in the latter part of this section.

Lemma 11 (see [14].)Let , , C₁ ∈ ℂ^q×m, and be given. Let , , , M = R_AB^*, N = A^*L_B, and S = B^*L_M. Then the following statements are equivalent:

(1)
equation (5) is consistent,
(2)

(68)
(3)

(69)

In this case, the general solution of (5) can be expressed as

(70)

where U₁, U₂, V₁, V₂, W₁, and W₂ are arbitrary matrices over ℂ with appropriate sizes.

Now we give the main theorem of this section.

Theorem 12. Let A_i, C_i, (i = 1,2, 3), B_j, and D_j, (j = 1,3) be given. Set

(71)

(72)

(73)

Then the following statements are equivalent:

(1)
system (6) is consistent,
(2)
the equalities in (14) and (17) hold, and
(74)
(3)
the equalities in (15) and (18) hold, and
(75)

In this case, the general solution of system (6) can be expressed as

(76)

where

(77)

where U₁, U₂, V₁, V₂, W₁, and W₂ are arbitrary matrices over ℂ with appropriate sizes.

Proof. (2) ⇔ (3): Applying Lemma 3 and Lemma 11 gives

(78)

By a similar approach, we can obtain that

(79)

(1) ⇔ (2): We separate the four equations in system (6) into three groups:

(80)

(81)

(82)

By Lemma 1, we obtain that system (80) is solvable if and only if (14), (81) is consistent if and only if (17). The general solutions to system (80) and (81) can be expressed as (16) and (19), respectively. Substituting (16) and (19) into (82) yields

(83)

Hence, the system (5) is consistent if and only if (80), (81), and (83) are consistent, respectively. It follows from Lemma 11 that (83) is solvable if and only if

(84)

We know by Lemma 11 that the general solution of (83) can be expressed as (77).

In Theorem 12, let A₁ and D₁ vanish. Then we can obtain the general solution to the following system:

(85)

Corollary 13. Let A₂, C₂, B₁, D₁, B₃, C₃, D₃, and be given. Set

(86)

Then the following statements are equivalent:

(1)
system (85) is consistent
(2)

(87)

(88)
(3)

(89)

In this case, the general solution of system (6) can be expressed as

(90)

where

(91)

where U₁, U₂, V₁, V₂, W₁, and W₂ are arbitrary matrices over ℂ with appropriate sizes.

Acknowledgments

The authors would like to thank Dr. Mohamed, Dr. Sivakumar, and a referee very much for their valuable suggestions and comments, which resulted in a great improvement of the original paper. This research was supported by the Grants from the National Natural Science Foundation of China (NSFC (11161008)) and Doctoral Program Fund of Ministry of Education of P.R.China (20115201110002).

References

1 Zhang H., Rank equalities for Moore-Penrose inverse and Drazin inverse over quaternion, Annals of Functional Analysis. (2012) 3, no. 2, 115–127, MR2948392, ZBL1255.15006.
10.15352/afa/1399899936
Web of Science® Google Scholar
2 Bao L., Lin Y., and Wei Y., A new projection method for solving large Sylvester equations, Applied Numerical Mathematics. (2007) 57, no. 5–7, 521–532, https://doi.org/10.1016/j.apnum.2006.07.005, MR2322428, ZBL1118.65028.
10.1016/j.apnum.2006.07.005
Web of Science® Google Scholar
3 Dehghan M. and Hajarian M., On the generalized reflexive and anti-reflexive solutions to a system of matrix equations, Linear Algebra and Its Applications. (2012) 437, no. 11, 2793–2812, https://doi.org/10.1016/j.laa.2012.07.004, MR2964725, ZBL1255.15019.
10.1016/j.laa.2012.07.004
Web of Science® Google Scholar
4 Farid F. O., Moslehian M. S., Wang Q.-W., and Wu Z.-C., On the Hermitian solutions to a system of adjointable operator equations, Linear Algebra and Its Applications. (2012) 437, no. 7, 1854–1891, https://doi.org/10.1016/j.laa.2012.05.012, MR2946365.
10.1016/j.laa.2012.05.012
Web of Science® Google Scholar
5 He Z. H. and Wang Q. W., A real quaternion matrix equation with with applications, Linear Multilinear Algebra. (2013) 61, 725–740.
10.1080/03081087.2012.703192
Web of Science® Google Scholar
6 Wang Q.-W. and Wu Z.-C., Common Hermitian solutions to some operator equations on Hilbert C^x-modules, Linear Algebra and Its Applications. (2010) 432, no. 12, 3159–3171, https://doi.org/10.1016/j.laa.2010.01.015, MR2639276.
10.1016/j.laa.2010.01.015
Web of Science® Google Scholar
7 Wang Q.-W. and Dong C.-Z., Positive solutions to a system of adjointable operator equations over Hilbert C^x-modules, Linear Algebra and Its Applications. (2010) 433, no. 7, 1481–1489, https://doi.org/10.1016/j.laa.2010.05.023, MR2680273.
10.1016/j.laa.2010.05.023
Web of Science® Google Scholar
8 Wang Q.-W., van der Woude J. W., and Chang H.-X., A system of real quaternion matrix equations with applications, Linear Algebra and Its Applications. (2009) 431, no. 12, 2291–2303, https://doi.org/10.1016/j.laa.2009.02.010, MR2563022, ZBL1180.15019.
10.1016/j.laa.2009.02.010
Web of Science® Google Scholar
9 Wang Q. W., Zhang H.-S., and Song G.-J., A new solvable condition for a pair of generalized Sylvester equations, Electronic Journal of Linear Algebra. (2009) 18, 289–301, MR2519916, ZBL1190.15019.
10.13001/1081-3810.1314
Web of Science® Google Scholar
10 Braden H. W., The equations A^TX ± X^TA = B, SIAM Journal on Matrix Analysis and Applications. (1999) 20, no. 2, 295–302, https://doi.org/10.1137/S0895479897323270, MR1646852.
10.1137/S0895479897323270
Web of Science® Google Scholar
11 Djordjević D. S., Explicit solution of the operator equation A^*X + X^±A = B, Journal of Computational and Applied Mathematics. (2007) 200, no. 2, 701–704, https://doi.org/10.1016/j.cam.2006.01.023, MR2289243, ZBL1113.47011.
10.1016/j.cam.2006.01.023
Web of Science® Google Scholar
12 Cao W., Solvability of a quaternion matrix equation, Applied Mathematics. (2002) 17, no. 4, 490–498, https://doi.org/10.1007/s11766-996-0015-2, MR1942843, ZBL1020.15014.
10.1007/s11766-996-0015-2
Google Scholar
13 Xu Q., Sheng L., and Gu Y., The solutions to some operator equations, Linear Algebra and Its Applications. (2008) 429, no. 8-9, 1997–2024, https://doi.org/10.1016/j.laa.2008.05.034, MR2446636, ZBL1147.47014.
10.1016/j.laa.2008.05.034
Web of Science® Google Scholar
14 Wang Q.-W. and He Z.-H., Some matrix equations with applications, Linear and Multilinear Algebra. (2012) 60, no. 11-12, 1327–1353, https://doi.org/10.1080/03081087.2011.648635, MR2989766.
10.1080/03081087.2011.648635
Web of Science® Google Scholar
15 Tian Y., Maximization and minimization of the rank and inertia of the Hermitian matrix expression A − BX − (BX) ^x, Linear Algebra and Its Applications. (2011) 434, no. 10, 2109–2139, https://doi.org/10.1016/j.laa.2010.12.010, MR2781681, ZBL1211.15022.
10.1016/j.laa.2010.12.010
Web of Science® Google Scholar
16 He Z.-H. and Wang Q.-W., Solutions to optimization problems on ranks and inertias of a matrix function with applications, Applied Mathematics and Computation. (2012) 219, no. 6, 2989–3001, https://doi.org/10.1016/j.amc.2012.09.024, MR2991998.
10.1016/j.amc.2012.09.024
Web of Science® Google Scholar
17 Liu Y. and Tian Y., Max-min problems on the ranks and inertias of the matrix expressions A − BXC ± (BXC) ^*, Journal of Optimization Theory and Applications. (2011) 148, no. 3, 593–622, https://doi.org/10.1007/s10957-010-9760-8, MR2769881, ZBL1223.90077.
10.1007/s10957-010-9760-8
Web of Science® Google Scholar
18 Chu D., Hung Y. S., and Woerdeman H. J., Inertia and rank characterizations of some matrix expressions, SIAM Journal on Matrix Analysis and Applications. (2009) 31, no. 3, 1187–1226, https://doi.org/10.1137/080712945, MR2558819.
10.1137/080712945
Web of Science® Google Scholar
19 Liu Y. and Tian Y., A simultaneous decomposition of a matrix triplet with applications, Numerical Linear Algebra with Applications. (2011) 18, no. 1, 69–85, https://doi.org/10.1002/nla.701, MR2769034, ZBL1249.15020.
10.1002/nla.701
Web of Science® Google Scholar
20 Zhang X., Wang Q.-W., and Liu X., Inertias and ranks of some Hermitian matrix functions with applications, Central European Journal of Mathematics. (2012) 10, no. 1, 329–351, https://doi.org/10.2478/s11533-011-0117-9, MR2863801, ZBL1253.15050.
10.2478/s11533-011-0117-9
Google Scholar
21 Wang Q.-W., The general solution to a system of real quaternion matrix equations, Computers & Mathematics with Applications. (2005) 49, no. 5-6, 665–675, https://doi.org/10.1016/j.camwa.2004.12.002, MR2134962, ZBL1138.15004.
10.1016/j.camwa.2004.12.002
Web of Science® Google Scholar
22 Tian Y., Equalities and inequalities for inertias of Hermitian matrices with applications, Linear Algebra and Its Applications. (2010) 433, no. 1, 263–296, https://doi.org/10.1016/j.laa.2010.02.018, MR2645083, ZBL1205.15033.
10.1016/j.laa.2010.02.018
Web of Science® Google Scholar
23 Marsaglia G. and Styan G. P. H., Equalities and inequalities for ranks of matrices, Linear and Multilinear Algebra. (1974) 2, 269–292, MR0384840, https://doi.org/10.1080/03081087408817070.
10.1080/03081087408817070
Google Scholar

All articles

Solving Optimization Problems on Hermitian Matrix Functions with Applications

Abstract

1. Introduction

2. Extremal Ranks and Inertias of Hermitian Matrix Function (10) with Some Restrictions

3. Relations between the Hermitian Part of the Solutions to (13) and (11)

4. The Solvability Conditions and the General Solution to System (6)

Acknowledgments

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Solving Optimization Problems on Hermitian Matrix Functions with Applications

Abstract

1. Introduction

2. Extremal Ranks and Inertias of Hermitian Matrix Function (10) with Some Restrictions

3. Relations between the Hermitian Part of the Solutions to (13) and (11)

4. The Solvability Conditions and the General Solution to System (6)

Acknowledgments

References

References

Related

Information