The recursive inverse eigenvalue problem (Q2706264)

The following recursive inverse eigenvalue problem of order \(n\) (\(\mathbf{RIEP}(n)\)) is investigated: Let \(F\) be a field, let \(s_1,\dots,s_n\in F\), and \(l_1,r_1\in F^{1\times 1}\), \(l_2,r_2\in F^{2\times 2},\dots,l_n,r_n\in F^{n\times n}\). Construct a matrix \(A\in F^{n\times n}\) such that \(l_i^{\top}A\langle i\rangle=s_il_i^{\top}\), \(A\langle i\rangle r_i=s_ir_i\), \(i=1,\dots,n\), where \(A\langle i\rangle\) denotes the \(i\)th leading principal submatrix of \(A\). NEWLINENEWLINENEWLINEFirst, the problem is solved in the general case, i.e. it is proved that a solution of \(\mathbf{RIEP}(n)\) exists if and only if there exists a solution of \(\mathbf{RIEP}(n-1)\) satisfying some natural conditions. A nonrecursive necessary and sufficient condition for the existence of a unique solution of \(\mathbf{RIEP}(n)\) is also obtained. NEWLINENEWLINENEWLINEThen these results are used to give solutions in some special classes of matrices: nonnegative matrices (using also Perron-Frobenius theory), \(Z\)-matrices, real symmetric matrices, positive semidefinite matrices and Stieltjes matrices.