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

Let image be a nontrivial involution; i.e., R=R−1≠±I. We say that image is R-symmetric (R-skew symmetric) if RAR=A (RAR=−A). Let image be one of the following subsets of image: (i) R-symmetric matrices; (ii) Hermitian R-symmetric matrices; (iii) R-skew symmetric matrices; (iv) Hermitian R-skew symmetric matrices. Let image with rank(Z)=m and Λ=diag(λ1,…,λm). The inverse eigenproblem consists of finding (Z,Λ) such that the set image is nonempty, and to find the general form of image. In all cases we use the special spectral properties of image to essentially characterize the set of admissible pairs (Z,Λ), and the special structure of the members of image to obtain the general solution of the inverse eigenproblem. Given an arbitrary image, the approximation problem consists of finding the unique matrix image 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).