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Irwing Kaplansky

Toeplitz and Hessenberg matrices in the problem of semiscalar equivalence of second order polynomial matrices



Bogdan Shavarovskii

Department of Algebra
Pidstryhach Institute for Applied Problems
of Mechanics and Mathematics
National Academy of Sciences of Ukraine




We consider the problem of determining whether two polynomial matrices can be transformed one to the one by multiplying from the left by nonsingular numerical matrix and from the right by invertible polynomial matrix. This equivalence relation is known as semiscalar equivalence. Large difficulties in this problem arise already for 2-by-2 matrices. In this report the semiscalar equivalence of polynomial matrices of second order is investigated. In particular, necessary and sufficient conditions are found under which two matrices of second order are semiscalarly equivalent. The main result is stated in the terms of determinants of Toeplitz and Hessenberg matrices



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