An inverse eigenproblem and an associated approximation problem for generalized reflexive and anti-reflexive matrices

✍ Scribed by Guang-Xin Huang; Feng Yin

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 230 KB
• Volume: 235
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2010.12.016

✦ Synopsis

In this paper, we first give the existence of and the general expression for the solution to an inverse eigenproblem defined as follows: given a set of real n-vectors {x i } m i=1 and a set of real numbers {λ i } m i=1 , and an n-by-n real generalized reflexive matrix A (or generalized antireflexive matrix B) such that {x i } m i=1 and {λ i } m i=1 are the eigenvectors and eigenvalues of A (or B), respectively, we solve the best approximation problem for the inverse eigenproblem.

That is, given an arbitrary real n-by-n matrix Ã, we find a matrix A à which is the solution to the inverse eigenproblem such that the distance between à and A à is minimized in the Frobenius norm. We give an explicit solution and a numerical algorithm for the best approximation problem over generalized reflexive (or generalized anti-reflexive) matrices.

Two numerical examples are also presented to show that our method is effective.