Cayley-Hamilton theorem for two variants of matrix spectral problems on E.Shmidt and development of characteristic polynomial.
A. N. Kuvshinova1, B. V. Loginov2
| Annotation | At the beginning of previous century E.Schmidt had introduced for integral operators eigenvalue systems $\left\{ {\lambda _k } \right\}$, counting with their multiplicities and relevant sets of eigenelements $\left\{ {\varphi _k } \right\}_1^\infty$, $\left\{ {\psi _k } \right\}_1^\infty$, such that $A\varphi _k = \lambda _k \psi _k ,\;A^* \psi _k = \lambda _k \varphi _k$. In this article generalized matrix spectral problems polynomially depending on Schmidt's spectral parameter are considered. I.S. Arjanykh (1951) has proved the generalized Hamilton-Cayley theorem for polynomial matrices with identity matrix at the parameter highest degree with the aim of application to numerical methods of linear algebra. Below it is given the extension of Hamilton-Cayley theorem on matrix E.Schmidt spectral problems polynomially depending on spectral parameter with identity matrix at the highest degree of spectral parameter (s.2) and also with identity (invertible) matrix at the parameter zero degree (s.3). With the aimes of further investigations \citetire{kuvshinovab11}{kuvshinovab13} on the base of the suggested by I.S.Arzhanykh \citetwo{kuvshinovab6}{kuvshinovab7} approach the development of characteristic polynomial on Schmidt spectral parameter degrees is made. |
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| Keywords | E.Schmidt spectrum; E.Shmidt eigenvalues; polynomial matrices on E.Shmidt spectral parameter; Hamilton-Cayley theorem; development of characteristic polynomial. |
1Post-graduate student of Higher Mathematics Chair, Ulyanovsk State Technical University, Ulyanovsk; erasya7@rambler.ru.
2Professor of Higher Mathematics Chair, Ulyanovsk State Technical University, Ulyanovsk; bvllbv@yandex.ru.
Citation: A. N. Kuvshinova, B. V. Loginov, "[Cayley-Hamilton theorem for two variants of matrix spectral problems on E.Shmidt and development of characteristic polynomial.]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,16:3 (2014) 7–20 (In Russian)
