Submatrix Constraint Research Articles

Inverse eigenvalue problems for bisymmetric matrices under a central principal submatrix constraint

This article considers an inverse eigenvalue problem for bisymmetric matrices under a central principal submatrix constraint and the corresponding optimal approximation problem. We first discuss the specified structure of bisymmetric matrices and their central principal submatrices. Then we study a special form for the matrix of independent eigenvectors for a bisymmetric matrix. Based on these, we give some necessary and sufficient conditions for the solvability of the inverse eigenvalue problem, and we derive an expression for its general solution. Finally, we obtain an expression for the solution to the corresponding optimal approximation problem.

Numerical solutions of AXB = C for mirrorsymmetric matrix X under a specified submatrix constraint

Mirrorsymmetric matrices, which are the iteraction matrices of mirrorsymmetric structures, have important application in studying odd/even-mode decomposition of symmetric multiconductor transmission lines (MTL). In this paper we present an efficient algorithm for minimizing $${\|AXB-C\|}$$where $${\|\cdot\|}$$is the Frobenius norm, $${A\in \mathbb{R}^{m\times n}}$$, $${B\in \mathbb{R}^{n\times s}}$$, $${C\in \mathbb{R}^{m\times s}}$$and $${X\in \mathbb{R}^{n\times n}}$$is mirrorsymmetric with a specified central submatrix [x ij ]r≤i, j≤n-r . Our algorithm produces a suitable X such that AXB = C in finitely many steps, if such an X exists. We show that the algorithm is stable any case, and we give results of numerical experiments that support this claim.

Inverse problem for symmetric $P$-symmetric matrices with a submatrix constraint

For a fixed generalized reflection matrix $P$, i.e., $P^T=P, P^2= I$, and $P\neq \pm I$, then a matrix $A $ is called a symmetric $P$-symmetric matrix if $A=A^T$ and $(PA)^T=PA$. This paper is mainly concerned with finding the least squares symmetric $P$-symmetric solutions to the matrix inverse problem $AX=B$ with a submatrix constraint, where $X$ and $B$ are given matrices of suitable size. By applying the generalized singular value decomposition and the canonical correlation decomposition, an analytical expression of the least squares solutions is derived basing on the Projection Theorem in Hilbert inner products spaces. Moreover, in the corresponding solution set, the analytical expression of the unique minimum-norm solution is described in detail.

The nearness problems for symmetric centrosymmetric with a special submatrix constraint

We say that \(X=[x_{ij}]_{i,j=1}^n\) is symmetric centrosymmetric if xij = xji and xn − j + 1,n − i + 1, 1 ≤ i,j ≤ n. In this paper we present an efficient algorithm for minimizing ||AXAT − B|| where ||·|| is the Frobenius norm, A ∈ ℝm×n, B ∈ ℝm×m and X ∈ ℝn×n is symmetric centrosymmetric with a specified central submatrix [xij]p ≤ i,j ≤ n − p. Our algorithm produces a suitable X such that AXAT = B in finitely many steps, if such an X exists. We show that the algorithm is stable any case, and we give results of numerical experiments that support this claim.

The submatrix constraint problem of matrix equation [formula omitted

We say that X = [ x ij ] i , j = 1 n is symmetric centrosymmetric if x ij = x ji and x n - j + 1 , n - i + 1 , 1 ⩽ i , j ⩽ n . In this paper we present an efficient algorithm for minimizing ‖ AXB + CYD - E ‖ where ‖ · ‖ is the Frobenius norm, A ∈ R t × n , B ∈ R n × s , C ∈ R t × m , D ∈ R m × s , E ∈ R t × s and X ∈ R n × n is symmetric centrosymmetric with a specified central submatrix [ x ij ] r ⩽ i , j ⩽ n - r , Y ∈ R m × m is symmetric with a specified central submatrix [ y ij ] 1 ⩽ i , j ⩽ p . Our algorithm produces suitable X and Y such that AXB + CYD = E in finitely many steps, if such X and Y exists. We show that the algorithm is stable any case, and we give results of numerical experiments that support this claim.

The inverse eigenvalue problem of reflexive matrices with a submatrix constraint and its approximation

In this paper, we first consider the existence of and the general expression for the solution to the constrained inverse eigenvalue problem defined as follows: given a generalized reflection matrix P∈Rn×n, a set of complex n-vectors {xi}i=1m, a set of complex numbers {λi}i=1m, and an s-by-s real matrix C0, find an n-by-n real reflexive matrix C such that the s-by-s leading principal submatrix of C is C0, and {xi}i=1m and {λi}i=1m are the eigenvectors and eigenvalues of C, respectively. We are then concerned with the best approximation problem for the constrained inverse problem whose solution set is nonempty. That is, given an arbitrary real n-by-n matrix \(\tilde{C}\) , find a matrix C which is the solution to the constrained inverse problem such that the distance between C and \(\tilde{C}\) is minimized in the Frobenius norm. We give an explicit solution and a numerical algorithm to the best approximation problem. An illustrative experiment is also presented.

Least squares solutions to AX = B for bisymmetric matrices under a central principal submatrix constraint and the optimal approximation

A matrix A ∈ R n × n is called a bisymmetric matrix if its elements a i , j satisfy the properties a i , j = a j , i and a i , j = a n - j + 1 , n - i + 1 for 1 ⩽ i , j ⩽ n . This paper considers least squares solutions to the matrix equation AX = B for A under a central principal submatrix constraint and the optimal approximation. A central principal submatrix is a submatrix obtained by deleting the same number of rows and columns in edges of a given matrix. We first discuss the specified structure of bisymmetric matrices and their central principal submatrices. Then we give some necessary and sufficient conditions for the solvability of the least squares problem, and derive the general representation of the solutions. Moreover, we also obtain the expression of the solution to the corresponding optimal approximation problem.

The nearness problems for symmetric matrix with a submatrix constraint

In this paper, we first give the representation of the general solution of the following least-squares problem (LSP): given a matrix X ∈ R n × p and symmetric matrices B ∈ R p × p , A 0 ∈ R r × r , find an n × n symmetric matrix A such that ∥ X T AX - B ∥ = min , s.t. A ( [ 1 , r ] ) = A 0 , where A ( [ 1 , r ] ) is the r × r leading principal submatrix of the matrix A. We then consider a best approximation problem: given an n × n symmetric matrix A ˜ with A ˜ ( [ 1 , r ] ) = A 0 , find A ^ ∈ S E such that ∥ A ˜ - A ^ ∥ = min A ∈ S E ∥ A ˜ - A ∥ , where S E is the solution set of LSP. We show that the best approximation solution A ^ is unique and derive an explicit formula for it.

Least‐squares solutions of matrix inverse problem for bi‐symmetric matrices with a submatrix constraint

AbstractAn n × n real matrix A = (aij)n × n is called bi‐symmetric matrix if A is both symmetric and per‐symmetric, that is, aij = aji and aij = an+1−1,n+1−i (i, j = 1, 2,..., n). This paper is mainly concerned with finding the least‐squares bi‐symmetric solutions of matrix inverse problem AX = B with a submatrix constraint, where X and B are given matrices of suitable sizes. Moreover, in the corresponding solution set, the analytical expression of the optimal approximation solution to a given matrix A* is derived. A direct method for finding the optimal approximation solution is described in detail, and three numerical examples are provided to show the validity of our algorithm. Copyright © 2007 John Wiley & Sons, Ltd.

Inverse problems for nonsymmetric matrices with a submatrix constraint

In this paper the following problems are considered: Problem I(a): Given matrices X ∈ R n × p with full column rank, B ∈ R p × p and A 0 ∈ R r × r , find a matrix A ∈ R n × n such that X T AX = B , A ( [ 1 , r ] ) = A 0 , where A ( [ 1 , r ] ) is the r × r leading principal submatrix of the matrix A. Problem I(b): Given matrices X ∈ R n × p , B ∈ R p × p and A 0 ∈ R r × r , find a matrix A ∈ R n × n such that ‖ X T AX - B ‖ = min s.t. A ( [ 1 , r ] ) = A 0 . Problem II: Given a matrix A ∼ ∈ R n × n with A ∼ ( [ 1 , r ] ) = A 0 , find A ^ ∈ S E such that ‖ A ∼ - A ^ ‖ = inf A ∈ S E ‖ A ∼ - A ‖ , where S E is the solution set of Problem I(a). By applying the generalized singular value decomposition (GSVD) and the canonical correlation decomposition (CCD) of a matrix pair, the solvability conditions for Problem I(a) and the general solution of Problem I are derived. The expression of the solution of Problem II is presented. A numerical algorithm for solving the problems is provided.

Inverse problems for symmetric matrices with a submatrix constraint

This paper is concerned with the following problems: Problem I(a). Given a full column rank matrix X ∈ R n × p and symmetric matrices B ∈ R p × p and A 0 ∈ R r × r , find an n × n symmetric matrix A such that X T A X = B , A ( [ 1 , r ] ) = A 0 , where A ( [ 1 , r ] ) is the r × r leading principal submatrix of the matrix A. Problem I(b). Given a matrix X ∈ R n × p and symmetric matrices B ∈ R p × p , A 0 ∈ R r × r , find an n × n symmetric matrix A such that ‖ X T A X − B ‖ = min , s.t. A ( [ 1 , r ] ) = A 0 . Problem II. Given an n × n symmetric matrix A ˜ , find A ˆ ∈ S E such that ‖ A ˜ − A ˆ ‖ = inf A ∈ S E ‖ A ˜ − A ‖ , where S E is the solution set of Problem I(a). By applying the generalized singular value decomposition (GSVD) and the canonical correlation decomposition (CCD) of a matrix pair, the solvability conditions for Problem I(a) and the general forms of the solution of Problem I are presented. The expression of the solution of Problem II is derived. A numerical algorithm for solving Problem II is provided.

The Inverse Eigenproblem of Centrosymmetric Matrices with a Submatrix Constraint and Its Approximation

In this paper, we first consider the existence of and the general expression for the solution to the constrained inverse eigenproblem defined as follows: given a set of complex $n$-vectors $\{\bx_i\}_{i=1}^{m}$ and a set of complex numbers $\{\la_i\}_{i=1}^m$, and an $s$-by-$s$ real matrix $C_0$, find an $n$-by-$n$ real centrosymmetric matrix $C$ such that the $s$-by-$s$ leading principal submatrix of $C$ is $C_0$, and $\{\bx_i\}_{i=1}^{m}$ and $\{\la_i\}_{i=1}^m$ are the eigenvectors and eigenvalues of $C$, respectively. We are then concerned with the best approximation problem for the constrained inverse problem whose solution set is nonempty. That is, given an arbitrary real $n$-by-$n$ matrix $\tilde{C}$, find a matrix $C$ which is the solution to the constrained inverse problem such that the distance between $C$ and $\tilde{C}$ is minimized in the Frobenius norm. We give an explicit solution and a numerical algorithm to the best approximation problem. Some illustrative experiments are also presented.

The inverse problem of bisymmetric matrices with a submatrix constraint

AbstractAn n×n real matrix A is called a bisymmetric matrix if A=AT and A=SnASn, where Sn is an n×n reverse unit matrix. This paper is mainly concerned with solving the following two problems:Problem IGiven n×m real matrices X and B, and an r×r real symmetric matrix A0, find an n×n bisymmetric matrix A such that where A([1: r]) is a r×r leading principal submatrix of the matrix A.Problem IIGiven an n×n real matrix A*, find an n×n matrix  in SE such that where ∥·∥ is Frobenius norm, and SE is the solution set of Problem I.The necessary and sufficient conditions for the existence of and the expressions for the general solutions of Problem I are given. The explicit solution, a numerical algorithm and a numerical example to Problem II are provided. Copyright © 2003 John Wiley & Sons, Ltd.

Classroom Note:Centrosymmetric Matrices

Attention is drawn to some earlier work on the odd--even properties of eigenvectors of centrosymmetric matrices.