The application of the rank-factorization method to the analysis of spectral characteristics of a polynomial multiparameter matrix

The method of rank-factorization (the $\Delta W$-$q$ method), previously suggested by the author as a method for solving multiparameter algebraic problems for matrix $F$ polynomially dependent on parameters, is applied to analyze the finite spectrum of the matrix $F$. Special attention is paid to the part of the spectrum $\sigma[F]$ of the $q$-parameter matrix $F$ whose points are independent of at least one of the spectral parameters.