Abstract
For a certain class of polynomial matrices A(x), we consider transformations S A(x) R(x) with invertible matrices S and R(x), i.e., the so-called semiscalarly equivalent transformations. We indicate necessary and sufficient conditions for this type of equivalence of matrices. We introduce the notion of quasidiagonal equivalence of numerical matrices. We establish the relationship between the semiscalar and quasidiagonal equivalences and the problem of matrix pairs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
P. S. Kazimirskii and V. M. Petrichkovich, “On the equivalence of polynomial matrices,” in: Theoretical and Applied Problems in Algebra and Differential Equations [in Ukrainian], Naukova Dumka, Kiev (1977), pp. 61–66.
P. S. Kazimirskii, Factorization of Matrix Polynomials [in Ukrainian], Naukova Dumka, Kiev (1981).
B. Z. Shavarovskii, “Semiscalar equivalence of polynomial matrices with pairwise different roots of their characteristic polynomial,” Mat. Met. Fiz.-Mekh. Polya, Issue 19, 33–37 (1984).
V. V. Sergeichuk and D. V. Galinskii, “Classification of pairs of linear operators in a four-dimensional vector space,” in: Infinite Groups and Related Algebraic Structures [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1993), pp. 413–430.
B. Z. Shavarovskii, “Quasidiagonal reduction of one class of polynomial matrices,” Mat. Met. Fiz.-Mekh. Polya, Issue 40, No.3, 7–12 (1997).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Shavarovskii, B.Z. On Semiscalar and Quasidiagonal Equivalences of Matrices. Ukrainian Mathematical Journal 52, 1638–1643 (2000). https://doi.org/10.1023/A:1010417504682
Issue Date:
DOI: https://doi.org/10.1023/A:1010417504682