A Class of Constrained Inverse Eigenproblem and Associated Approximation Problem for Symmetric Reflexive Matrices

Let S∈Rn×n be a symmetric and nontrival involution matrix. We say that A∈E R n×n is a symmetric reflexive matrix if AT = A and SAS = A. Let S R r n×n(S)={A|A= AT,A = SAS, A∈Rn×n}. This paper discusses the following two problems. The first one is as follows. Given Z∈Rn×m (m < n),∧= diag(λ1,...,λm)∈Rm×m, andα,β∈R withα<β. Find a subset (?)(Z,∧,α,β) of SRrn×n(S) such that AZ = Z∧holds for any A∈(?)(Z,∧,α,β) and the remaining eigenvaluesλm 1 ,...,λn of A are located in the interval α,β], Moreover, for a given B∈Rn×n, the second problem is to find AB∈(?)(Z,∧,α,β) such that where ||.|| is the Frobenius norm. Using the properties of symmetric reflexive matrices, the two problems are essentially decomposed into the same kind of subproblems for two real symmetric matrices with smaller dimensions, and then the expressions of the general solution for the two problems are derived.

A Class of Constrained Inverse Eigenproblem and Associated Approximation Problem for Symmetric Reflexive Matrices

Let S ∈ Rn×n be a symmetric and nontrival involution matrix. We say that A ∈ Rn×n is a symmetric reflexive matrix if AT = A and SAS = A. Let SRrn×n(S)={A|A =AT, A = SAS, A ∈ Rn×n}. This paper discusses the following two problems. The first one is as follows.Given Z∈Rn×m (m＜n),∧=diag(λ1…,λm)∈Rm×m,and α,β∈R with α＜β. Find a subset ψ(Z,A,α,β) of SRrn×n(S) such that AZ = ZA holds for any A∈ ψ(Z,∧,α, β)and the remaining eigenvalues λm+1,…λn of A are located in the interval α, β]. Moreover, for a given B ∈ Rn×n, the second problem is to find AB ∈ψ(Z, A, α,β)such that ||B-AB||= min A∈ψ(A,∧,α,β) ||B - A||,where ||.|| is the Frobenius norm. Using the properties of symmetric reflexive matrices,the two problems are essentially decomposed into the samekind of subproblems for two real symmetric matrices with smaller dimensions, and then the expressions of the general solution for the two problems are derived.