Inverse eigenproblem for R-symmetric matrices and their approximation

Authors:
Yongxin Yuan
Affiliations:
School of Mathematics and Physics, Jiangsu University of Science and Technology, Zhenjiang, 212003, PR China
Venue:
Journal of Computational and Applied Mathematics
Year:
2009

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Abstract

Let R@?C^n^x^n be a nontrivial involution, i.e., R=R^-^1+/-I"n. We say that G@?C^n^x^n is R-symmetric if RGR=G. The set of all nxnR-symmetric matrices is denoted by GSC^n^x^n. In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors {x"i}"i"="1^m in C^n and a set of complex numbers {@l"i}"i"="1^m, find a matrix A@?GSC^n^x^n such that {x"i}"i"="1^m and {@l"i}"i"="1^m are, respectively, the eigenvalues and eigenvectors of A. We then consider the following approximation problem: Given an nxn matrix A@?, find A@?@?S"E such that @?A@?-A@?@?=min"A"@?"S"""E@?A@?-A@?, where S"E is the solution set of IEP and @?@?@? is the Frobenius norm. We provide an explicit formula for the best approximation solution A@? by means of the canonical correlation decomposition.