To solving spectral problems for $q$-parameter polynomial matrices.~3

V. N. Kublanovskaya

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXV, Tome 405 (2012), pp. 164-169

Résumé

The paper suggests methods for computing points of the finite spectrum of a multiparameter matrix pencil (a multiparameter polynomial matrix linearly dependent on its parameters) of general form. At the first stage, a sequence $\{A_k+\mu_kB_k\}$ of pencils is computed, where $B_k$ are constant matrices and $A_k$ are $(q-k)$-parameter matrices linearly dependent on parameters, $k=1,\dots,q$. At every step of the second stage, which is different for the regular and singular spectra, an auxiliary one- or two-parameter hereditary pencil is formed, and the points of its spectrum are computed. In order to determine whether the characteristics computed belong to spectrum points of the original matrix, the hereditary pencils are used. Their construction is based on computing bases of null spaces of constant or one-parameter matrices.