An explicit formula for singular values of the Sylvester-Kac matrix (Q869910)

Let \(P(n)=(p_{ij}(n))\) be the \((n+1)\times (n+1)\) tridiagonal centrosymmetric matrix satisfying \(p_{i+1,i}(n)=i\) and \(p_{i,i}(n)=0\), for all \(i\). The authors show that, for all nonnegative integers \(m\), (i) the singular values of \(P(2m)\) include the numbers \(\sqrt{(2m+1)^2-(2i+1)^2}\), \(i=0,1,2,\dots ,m\), and (ii) the singular values of \(P(2m+1)\) all have multiplicity two. The authors first use the Perron-Frobenius canonical form to reduce the singular value problem for \(P(n)\) to the singular value problems for two smaller matrices, a step similar to the usual splitting technique for centrosymmetric matrices. They then find the singular values of one of these smaller matrices by finding all nontrivial polynomial solutions of a related differential equation. These polynomials provide the elements of the singular vectors. The paper also includes a brief discussion of the work of J. J. Sylvester, M. Kac and others on the eigenvalue problem for \(P(n)\).