{"title":"The Inverse Problem of Nonsymmetric Matrices with a Submatrix Constraint and its Approximation","authors":"Yongxin Yuan, Hao Liu","country":null,"institution":"","volume":43,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":861,"pagesEnd":866,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/3805","abstract":"

In this paper, we first give the representation of the general solution of the following least-squares problem (LSP): Given matrices X ∈ Rn×p, B ∈ Rp×p and A0 ∈ Rr×r, find a matrix A ∈ Rn×n such that XT AX − B = min, s. t. A([1, r]) = A0, where A([1, r]) is the r×r leading principal submatrix of the matrix A. We then consider a best approximation problem: given an n × n matrix A˜ with A˜([1, r]) = A0, find Aˆ ∈ SE such that A˜ − Aˆ = minA∈SE A˜ − A, where SE is the solution set of LSP. We show that the best approximation solution Aˆ is unique and derive an explicit formula for it. Keyw<\/p>\r\n","references":null,"publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 43, 2010"}