Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix

1 Introduction

Throughout this article we use some notations as follows. Let C ^n×m be the set of all n × m complex matrices, R ^n×m be the set of all n × m real matrices, C ⁿ = C ^n×1, R ⁿ = R ^n×1, R denote the set of all real numbers, OR ^n×n denote the set of all n × n orthogonal matrices, R(A), A ^T , r(A), tr(A) and A ⁺ be the column space, the transpose, rank, trace and the Moore–Penrose generalized inverse of a matrix A, respectively. I _n is the identity matrix of size n. Let e _i be the ith column of I _n , and set J _n = (e _n ,…,e ₁). For A, B ∈ R ^n×m , 〈A, B〉 = tr(B ^T A) denotes the inner product of matrices A and B. The induced matrix norm is called the Frobenius norm, i.e. | | A | | = 〈 A , A 〉 1 / 2 = ( tr ( A T A ) ) 1 / 2 , then R ^n×m is a Hilbert inner product space.

Generally, the left and right inverse eigenpairs problem is as follows: given partial left and right eigenpairs (eigenvalue and corresponding eigenvector) (γ _j , y _j ), j = 1,…,l; (λ _i , x _i ), i = 1,…,h, and a special n × m matrix set S, to find A ∈ S such that

where h ≤ n and l ≤ n. If X = (x ₁,…,x _h ), Λ = diag(λ ₁,…,λ _h ), Y = (y ₁,…,y _l ), Γ = diag(γ ₁,…,γ _l ), then (1.1) is equivalent to

This problem, which mainly arises in perturbation analysis of matrix eigenvalue and in recursive matters, has some practical applications in economic and scientific computation fields [1,2,3].

Many important results have been achieved on this problem associated with many kinds of matrix sets. Li et al. [4,5,6,7,8,9] have solved the left and right inverse eigenpairs problems for skew-centrosymmetric matrices, generalized centrosymmetric matrices, κ-persymmetric matrices, symmetrizable matrices, orthogonal matrices and κ-Hermitian matrices by using the special properties of eigenpairs of matrix. Zhang and Xie [10], Ouyang [11], Liang and Dia [12] and Yin and Huang [13] have, respectively, solved the left and right inverse eigenvalue problems for real matrices, semipositive subdefinite matrices, generalized reflexive and anti-reflexive matrices and (R,S) symmetric matrices with the special structure of matrix.

Arav et al. [2] and Loewy and Mehrmann [3] studied the recursive inverse eigenvalue problem which arises in the Leontief economic model. Namely, given eigenvalue λ _i of A _i , in which A _i is the ith leading principal submatrix of A, corresponding left eigenvector y _i and right eigenvector x _i of λ _i , construct a matrix A ∈ C ^n×m such that

This recursive inverse eigenvalue problem is a special case of the left and right inverse eigenvalue problem with the leading principal submatrix constraint. Few authors have considered the left and right inverse eigenpairs problem with a submatrix constraint. In this article, we will consider the left and right inverse eigenpairs problem with the leading principal submatrix constraint for the generalized centrosymmetric matrix, which has not been discussed.

From Definition 1, it is easy to derive the following conclusions.

1. J ^T = J and J ² = I _n . Real matrices and centrosymmetric matrices are the special cases of generalized centrosymmetric matrices with κ(i) = i and κ(i) = n − i + 1 or J = I _n and J = J _n , respectively.

2. A ∈ GCSR ^n×n if and only if A = JAJ and A ∈ GCSSR ^n×n if and only if A = −JAJ.

3. R ^n×n = GCSR ^n×n ⊕ GCSSR ^n×n , where the notation V ₁ ⊕ V ₂ stands for the orthogonal direct sum of linear subspaces V ₁ and V ₂.

Centrosymmetry, persymmetry and symmetry are three important symmetric structures of a square n × n matrix and have profound applications, such as engineering, statistics and so on [14,15,16]. There are many meaningful results about the inverse problem and the inverse eigenvalue problem of centrosymmetric matrices with a submatrix constraint. Peng et al. [17] and Bai [18] discussed the inverse problem and the inverse eigenvalue problem of centrosymmetric matrices with a principal submatrix constraint, respectively. Zhao et al. [19] studied least squares solutions to AX = B for symmetric centrosymmetric matrices under a central principal submatrix constraint and the optimal approximation. The matrix inverse problem (or inverse eigenvalue problem) with a submatrix constraint is also called the matrix extension problem. Since de Boor and Golub [20] first put forward and considered the Jacobi matrix extension problem in 1978, many authors have studied the matrix extension problem and a series of meaningful results have been achieved [17,18,19,21,22,23,24,25,26].

Assume (λ _i ,x _i ), i = 1,…,m, be right eigenpairs of A; (μ _i ,y _i ), j = 1,…,h, be left eigenpairs of A. Let X = (x ₁,…,x _m ) ∈ C ^n×m , Λ = diag(λ ₁,…,λ _m ) ∈ C ^m×m ; Y = (y ₁,…,y _h ) ∈ C ^n×h , Γ = diag(μ ₁,…,μ _h ) ∈ C ^h×h . The problems studied in this article can be described as follows.

where A[1:p] denotes the p × p leading principal submatrix.

This article is organized as follows. In Section 2, we first study the special properties of eigenpairs and the structure of generalized centrosymmetric matrices. Then, we provide the solvability conditions for and the general solutions of Problem I. In Section 3, we first attest the existence and uniqueness theorem of Problem II and then present the unique approximation solution with the orthogonal invariance of the Frobenius norm. Finally, we provide an algorithm to compute the unique approximation solution. Some conclusions are provided in Section 4.

2 Solvability conditions of Problem I

where A 11 = K 1 T A K 1 ∈ R ( n − r ) × ( n − r ) , A 22 = K 2 T A K 2 ∈ R r × r .

If J = J _n and n = 2k, then

If J = J _n , and n = 2k + 1, then

Similarly, we have the following splitting of centrosymmetric matrices into smaller submatrices using K.

For a real matrix A ∈ R ^n×m , its complex right eigenpairs are conjugate pairs. That is, if a + b − 1 and x + − 1 y are one of its right eigenpairs, then a − b − 1 and x − − 1 y are one of its right eigenpairs. This implies Ax = ax − by and Ay = bx + ay, i.e.,

Therefore, in Problem I, we can assume that X ∈ R ^n×m and

where λ ¯ i , i = 1 , … , m ¯ are real numbers or 2 × 2 real matrices, m ¯ ≤ m . m ¯ = m holds if and only if all right eigenvalues of A are real numbers. We can also prove that the complex left eigenpairs of A are conjugate pairs. Hence, in Problem I, we can also assume that Y ∈ R ^n×h and

where μ ¯ i , i = 1 , … , h ¯ are real numbers or 2 × 2 real matrices, h ¯ ≤ h . h ¯ = h holds if and only if all left eigenvalues of A are real numbers.

Let A ∈ GCSR ^n×n , if Ax = λx, where λ is a number, x ∈ C ⁿ , and x ≠ 0, then we have

Hence, we have A(x ± Jx) = λ(x ± Jx). It is obvious that x + Jx and x − Jx is a generalized symmetric vector and a generalized skew-symmetric vector, respectively. If a + b − 1 and x + − 1 y are one of its right eigenpairs, then we have

According to conclusion (2) of Definition 1, we have

Combining (2.3) and (2.4) implies

It is easy to see that the columns of (x; y) + J(x; y) (or (x; y) − J(x; y)) is a generalized symmetric vector (or generalized skew-symmetric vector). Hence, the right eigenvectors of the generalized centrosymmetric matrix can be expressed as generalized symmetric vectors or generalized skew-symmetric vectors. It is clear that the left eigenvectors of the generalized centrosymmetric matrix have the same properties as the right ones. According to the aforementioned analysis, in Problem I, we may assume as follows:

Combining Lemmas 1 and 3, it is easy to prove that A in (2.8) can be expressed as

where A ₁₁₀, A ₂₁₀ are denoted by A ₁₀, E ₁, E ₂ are denoted by E, G ₁, G ₂ are denoted by G, and A ₁₁₀, E ₁, G ₁ ∈ R ^{(n−k)×(n−k)}, A ₂₁₀, E ₂, G ₂ ∈ R ^k×k , for any F ₁ ∈ R ^{(n−k)×(n−k)}, F ₂ ∈ R ^k×k .

Denote

Combining (2.11) and (2.12), A[1:p] = A ₀ if and only if the following equation holds.

Suppose that the generalized singular value decomposition (GSVD) of matrix pairs ( E ¯ 1 T , E ¯ 2 T ) is as follows:

where Q ₁ ∈ OR ^{(n−k)×(n−k)}, Q ₂ ∈ OR ^k×k , S ∈ R ^p×p is nonsingular, and

with s 1 = r ( E ¯ 1 , E ¯ 2 ) T , r 1 = s 1 − r ( E ¯ 2 T ) , t 1 = r ( E ¯ 1 T ) + r ( E ¯ 2 T ) − s 1 , Θ 1 = diag ( γ 1 , … , γ t 1 ) , Θ 2 = diag ( δ 1 , ⋯ , δ t 1 ) , with 1 ≥ γ t 1 ≥ ⋯ ≥ γ 1 > 0 , 0 < δ 1 ≤ ⋯ ≤ δ t 1 ≤ 1 , γ i 2 + δ i 2 = 1 , i = 1 , … , t 1 .

Suppose that the GSVD of matrix pairs ( G ¯ 1 , G ¯ 2 ) is

where P ₁ ∈ OR ^{(n−k)×(n−k)}, P ₂ ∈ OR ^k×k , W ∈ R ^p×p is nonsingular, and

with s 2 = r ( G ¯ 1 G ¯ 2 ) , r 2 = s 2 − r ( G ¯ 2 ) , t 2 = r ( G ¯ 1 ) + r ( G ¯ 2 ) − s 2 , Θ 3 = diag ( α 1 , … , α t 2 ) , Θ 4 = diag ( β 1 , … , β t 2 ) , with 1 ≥ α t 2 ≥ ⋯ ≥ α 1 > 0 , 0 < β 1 ≤ ⋯ ≤ β t 2 ≤ 1 , α i 2 + β i 2 = 1 , i = 1 , … , t 2 , where r ₁, t ₁, s ₁ − r ₁ − t ₁, p − s ₁, r ₂, t ₂, s ₂ − r ₂ − t ₂, p − s ₂ denote the number of columns of the sub-block-matrix of Σ ₁, Σ ₂, Σ ₃ and Σ ₄.

Combining (2.14) and (2.16) implies that (2.13) can be written as

Partition Q 1 T F 1 P 1 , Q 2 T F 2 P 2 , S − 1 A ¯ 0 W − T into the following form:

Combining (2.19) and (2.20) implies that (2.18) can be written as

Combining Lemma 3 and (2.11)–(2.21) derives the following theorem.

3 An expression of the solution of Problem II

From (2.23), it is easy to prove that the solution set S _E of Problem I is a nonempty closed convex set if Problem I has a solution in GCSR ^n×n . We claim that for any given A* ∈ R ^n×n , there exists a unique optimal approximation for Problem II.

Combining (2.8)–(2.11) and Lemma 1, (2.23) can be written as

where E and G are denoted by (2.10), F ₁ and F ₂ are denoted by (2.24) and (2.25), respectively.

According to conclusion (3) of Definition 1, for any A* ∈ R ^n×n , there exist only one A 1 ⁎ ∈ G C S R n × n and only one A 2 ⁎ ∈ G C S S R n × n which satisfy

where

According to Lemma 1, A 1 ⁎ can be written as

where A 11 ⁎ ∈ R ( n − k ) × ( n − k ) , A 22 ⁎ ∈ R n × n are given by A 1 ⁎ . Denote

According to (2.10), it is easy to prove that E, F are orthogonal projection matrices. Hence, there exist orthogonal projection matrices E ¯ , G ¯ which satisfy

From this, we have

This implies that

According to (2.9) and (2.10), it is easy to prove EA ₁₀ G = 0. Hence, we have

It is clear that if F ˆ = A 1 ⁎ + E ¯ F ¯ G ¯ , for any F ¯ ∈ G C S R n × n , then

Combining Lemma 1, (2.11) and (3.10), we have

where E ₁, E ₂, G ₁, G ₂ are denoted by (2.11).

Denote

Combining (3.1) and (3.5), we have