On Doubly Symmetric Tridiagonal Forms for Complex Matrices and Tridiagonal Inverse Eigenvalue Problems
Alan D. George, Х. Д. Икрамов, Wen-Ling Tang, V. N. Tchugunov · SIAM Journal on Matrix Analysis and Applications · 1996
It is well known that for any distinct real numbers $\lambda _1 , \cdots ,\lambda _n $ there exists a doubly symmetric (i.e., symmetric and persymmetric or, equivalently, symmetric and centrosymmetric) tridiagonal real $n \times n$ matrix T with the $\lambda $’s as its eigenvalues. Such a matrix can be constructed finitely using only arithmetic operations and square roots. We prove in this paper that the analogous assertions hold for any distinct complex numbers $\lambda _1 , \cdots ,\lambda _n $ with T being a complex matrix. It follows that any complex $n \times n$ matrix with distinct eigenvalues is similar to a doubly symmetric tridiagonal matrix. The condition that the eigenvalues be distinct is essential: we show that the tridiagonal form above does not exist or is trivial (depending on the Jordan structure imposed) if $\lambda _1 = \lambda _2 = \lambda _3 = 0$.