On a class of inverse palindromic eigenvalue problem

In this paper we first consider the following inverse palindromic eigenvalue problem (IPEP): Given matrices Lambda = diag{lambda 1, ... lambda p} is an element of C-pxp, lambda(i) not equal lambda(j) for i, j, i; j = 1; ... ; p, X = [x(1); .; x(p)] is an element of C-nxp with rank(X) = p, and both Lambda and X are closed under complex conjugation in the sense that lambda(2i) = lambda(2i-1) is an element of C, x(2i) = x(2i-1) is an element of C-n for i = 1; ...; m, and lambda(j) is an element of R, x(j) is an element of R-n for j = 2m + 1; ... p, find a matrix A is an element of R-nxn such that AX = A(T)X Lambda: We then consider a best approximation problem (BAP): Given (A) over tilde is an element of R-nxn, find A is an element of S-A such that parallel to A-A parallel to = min(A is an element of SA) parallel to A-(A) over tilde parallel to where parallel to.parallel to is the Frobenius norm and SA is the solution set of IPEP. By partitioning the matrix Lambda and using the QR-decomposition, the expression of the general solution of Problem IPEP is derived. Also, we show that the best approximation solution A is unique and derive an explicit formula for it.