The common bisymmetric nonnegative definite solutions with extreme ranks and inertias to a pair of matrix equations

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Abstract

We in this paper consider the bisymmetric nonnegative definite solution with extremal ranks and inertias to a system of quaternion matrix equations AX = C, XB = D. We derive the extremal ranks and inertias of the common bisymmetric nonnegative definite solution to the system. The general expressions of the bisymmetric nonnegative definite solution with extremal ranks and inertias to the system mentioned above are also presented. In addition, we give a numerical example to illustrate the results of this paper.

Introduction

Throughout this paper, we denote the real number field by R, the set of all m × n matrices over the quaternion division algebraH=a0+a1i+a2j+a3k|i2=j2=k2=ijk=-1,a0,a1,a2,a3Rby Hm×n; the symbols I,A,A-,A,R(A),N(A) stand for the identity matrix with the appropriate size, the conjugate transpose, “an any but fixed” inner inverse and the Moore–Penrose inverse, the column right space, the row left space of a matrix A over H, respectively. Moreover, RA = I  AA, LA = I  AA, EA = I  AA and FA = I  AA. We use symbols dimR(A) and dimN(A) denote the dimension of R(A) and the dimension of N(A), respectively. For a quaternion matrix A,dimR(A)=dimN(A), which we call the rank of A and denote by r(A).

Let A=(aij)Hm×n,A=(aji¯)Hn×m,A#=(am-i+1,n-j+1¯)Hm×n, where aji¯ is the conjugate of the quaternion aji. The matrix A=(aij)Hn×n is called Hermitian if A = A. We denote Hhn×n by the sets of all n × n quaternion Hermitian matrices. The matrix A=(aij)Hn×n is called bisymmetric if A = A = A# [13]. We denote the set of all n × n bisymmetric matrices by Bn. If ABn is nonnegative definite, we call A the bisymmetric nonnegative definite matrix.

It is well-known that the eigenvalues of a Hermitian matrix AHn×n are all real and the inertia of A is defined to be the tripletIn(A)=i+(A),i-(A),i0(A),where i+(A), i(A) and i0(A) denote 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.

Bisymmetric and bisymmetric nonnegative definite matrices have wide applications in engineering problem, information theory, linear system theory and numerical analysis. Some important results on bisymmetric and bisymmetric nonnegative definite solutions to some matrix equations can be found. For instance, Wang in [12] considered the bisymmetric solution to the system of matrix equationsAX=C,XB=D,over H. Recently, Zhang and Wang [14] investigated the (P, Q)-(skew) symmetric extremal rank solutions to system (1.1). In 2000, Xie et al. [4] presented the bisymmetric nonnegative definite solution to AX = B over R. In 2002, using the generalized singular value decomposition, Liao and Bai [1] presented the bisymmetric nonnegative definite solution X of the matrix equationDTXD=C,over R. In 2010, Dehghan and Hajarian [9] presented an iterative algorithm to get the common generalized bisymmetric matrix pair solution (X, Y) to the generalized coupled Sylvester matrix equations AXB + CYD = M, EXF + GYH = N. The other results concerned with bisymmetric solution to different matrix equations can also be found in ([8], [10], [32], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47]).

As we known, ranks of solutions of linear matrix equations have been considered previously by several authors (see, e.g. [11], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31]). Recently, Xiao et al. [16] investigated the symmetric minimal rank solution to the matrix equation AX = B. However, to our knowledge, there has been little information on bisymmetric nonnegative definite solutions with extremal ranks and inertias to the system (1.1). This paper aims to discuss this problem. For the more relevant work, we can see the literature (e.g. [2], [5], [6], [35], [36]).

We organize this paper as follows. In Section 2, we give the general expressions of the bisymmetric nonnegative definite solution to system (1.1) when the solvability conditions are satisfied. In Section 3, we give the general expression and ranks and inertias formulas for bisymmetric nonnegative definite solution which has the extremal ranks and inertias. In Section 4, we give a numerical example.

Section snippets

Bisymmetric nonnegative definite solution to (1.1)

This section presents the general expressions of the bisymmetric nonnegative definite solution to the system (1.1) over H. We begin with the following lemmas.

Lemma 2.1

See [12]

Assume that Vn is an n × n permutation matrix whose elements along the southwest–northeast diagonal are ones and whose remaining elements are zeros. Then:

  • (a)

    KB2r if and only ifK=Q-1X100X2Q-,where X1, X2 are r × r Hermitian matrices andQ=Ir-VrVrIr;

  • (b)

    KB2r+1 if and only if (2.1) holds, where X1, X2 are Hermitian matrices with respective size r × r, (r

The extremal ranks and inertias for the bisymmetric nonnegative definite solution to (1.1) over H

In this section, we derive the representation of the bisymmetric nonnegative definite solution to (1.1) over H with extremal ranks and inertias.

The following lemmas provide us some useful results about the ranks and inertias of matrices over C, which can be generalized to H.

Lemma 3.1

See [7]

Let AHm×n,BHm×k,CHl×n be given. Thenr(A,B)=r(A)+r(RAB)=r(B)+r(RBA),rAC=r(A)+r(CLA)=r(C)+r(ALC).

Lemma 3.2

See [34] or [33]

Let P(X) = A  BXB, where AHhm×m,BHm×n be given, and XHhn×n is unknown matrix, and letM=ABB0,S=0In,S1=S-MMS.ThenmaxXHhn×nr(P

A numerical example

In this section, we give a numerical example to illustrate the results of this paper.

We consider the bisymmetric nonnegative definite solutions with extremal rank and positive index of inertia to the system of quaternion matrix equationsAX=C,XB=D,whereA=1+jk-jk-j1-j-11-ii-11,B=2+2k2-1+3j-2k1-i+j-k1+j-2k-1+i+j-k2-2k0,C=6-2j+3k3+0.5i-1.5j+k3-0.5i-1.5j+2k6-4j+3k-1.5-0.5i-0.5j-0.5k1-1.5i-0.5k-1+1.5i+0.5k1.5+0.5i+0.5j+0.5k,D=12-i+6j-4k7.5+0.5i+2.5j-2.5k6+5j-3k4-1.5i+1.5j-2k6+j-3k2+1.5i+1.5j-k12+i+6j-

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    This research was supported by the Grants from Natural Science Foundation of Shanghai (11ZR1412500), the Ph.D. Programs Foundation of Ministry of Education of China (20093108110001), Natural Science Foundation of China (11171205), and Shanghai Leading Academic Discipline Project (J50101) and the Innovation Funds for Graduates of Shanghai University (SHUCX092005).

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