Inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry

Abstract

Let R ∈ C n×n be a nontrivial involution; i.e., R = R −1 / = ±I . We say that A ∈ C n×n is R-symmetric (R-skew symmetric) if RAR = A (RAR = −A). Let S be one of the following subsets of C n×n : (i) R-symmetric matrices; (ii) Hermitian R-symmetric matrices; (iii) Rskew symmetric matrices; (iv) Hermitian R-skew symmetric matrices. Let Z ∈ C n×m with rank(Z) = m and = diag(λ 1 , . . . , λ m ).The inverse eigenproblem consists of finding (Z, ) such that the set S(Z, ) = {A ∈ S|AZ = Z } is nonempty, and to find the general form of A ∈ S(Z, ). In all cases we use the special spectral properties of S to essentially characterize the set of admissible pairs (Z, ), and the special structure of the members of S to obtain the general solution of the inverse eigenproblem.Given an arbitrary B ∈ S, the approximation problem consists of finding the unique matrix A B ∈ S( , Z) that best approximates B in the Frobenius norm.It is not necessary to assume that R = R * in connection with the inverse eigenproblem for R-symmetric or R-skew symmetric matrices. However, we impose this additional assumption in connection with the inverse eigenproblem for Hermitian R-symmetric or R-skew symmetric matrices, and in connection with the approximation problem for (i)-(iv).

Reflexive solution to a system of matrix equations

“…Many researchers have given structure property of the generalized reflection matrix P ∈ C n×n with different methods (e.g, [7], [10]). By [7], for P 1 = 1 2 (I − P ) and P 2 = 1 2 (I + P ), there exist unit column orthogonal matrices U 1 ∈ C n×r , U 2 ∈ C n×(n−r) such that…”

“…Also reflexive and antireflexive matrices with respect to a generalized reflection matrix are very useful in engineering and scientific computations and have been extensively studied (see [7]- [10]). …”

Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems

“…The study of centrosymmetry has a long history [11][12][13][14][15][16][17][18][19][20]. However, the last two decades has stemmed much research focused on the properties and applications of centrosymmetric matrices ranging from iterative methods for solving linear equations to least-squares problems to inverse eigenvalue problems [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36].…”

An efficient method for solving a matrix least squares problem over a matrix inequality constraint

“…In particular, if we drop the constraint L ≤ E X F ≤ U and simplify the objection function in (1.1) as AX − B , then the Problem MLSMI is reduced to the Procrustes problems or the inverse eigenvalue problems with special structures, which have been widely investigated in the recent literature. See, for example, the symmetric case in [1][2][3][4], the centrosymmetric case in [5], the R symmetric cases in [6], the (R, S)-symmetric case in [7], the Re-nonnegative definitive case in [8], the reflexive and anti-reflexive case in [9], the symmetric orthogonal case in [10] and so on. On the other hand, if we simplify the constraint L ≤ E X F ≤ U in (1.1) to a nonnegative constraint X ≥ 0 or a bound constraint L ≤ X ≤ U , then the similar problems have been studied by using some projection-type methods.…”