On a nontrivial situation with pseudounitary eigenvalues of a positive definite matrix

Kh. D. Ikramov

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXV, Tome 514 (2022), pp. 55-60

Résumé

Let $I_{p,q} = I_p \oplus -I_q$. Pseudounitary eigenvalues of a positive definite matrix $A$ are the moduli of the conventional eigenvalues of the matrix $I_{p,q}A$. They are invariants of pseudounitary *-congruences performed with $A$. For a fixed $n = p + q$, the sum of the squares $\sigma_{p,q}$ of these numbers is a function of the parameter $p$. In general, its values for different $p$ can vary very significantly. However, if $A$ is the tridiagonal Toeplitz matrix with the entry $a \ge 2$ on the main diagonal and the entry $-1$ on the two neighboring diagonals, then $\sigma_{p,q}$ has a constant value for all $p$. This nontrivial fact is explained in the paper.