Least-square solutions for inverse problems of centrosymmetric matrices

Let <F>CSRnxn={A=(aij)∊Rnxn|aij=an+1−j, i,j=1,2…, n}</F> In this paper, we mainly discuss solving the following problems. PROBLEM Io Given <F>X, B ∊ Rnxm</F> Find <F>A ∊ CSRn×n</F> such that <F>AX = B</F>. PROBLEM I. Given <F>X, B ∊ Rn×m</F>. Find <F>A ∊ CSRn×n</F> such that <F>&par;AX−B&par;=min.</F>PROBLEM II. Given <F>A&grave; ∊ Rn×n</F>. Find <F>A&ast; ∊ SE</F> such that <INLINE-FIG BASELINE="0.0"><LINK LOCATOR="fx1"></INLINE-FIG> where &par; · &par; is the Frobenius norm, and SE is the solution set of Problem I or I0.The expressions of the general solution of Problems I and 10 have been derived. The sufficient and necessary conditions for the solvability of Problem I0 have been provided. The numerical algorithm to find the optimal approximate solution and some numerical experiments have been given. [Copyright &y& Elsevier]