# Shavarovskyy B. Z.

### Canonical form of polynomial matrices with all identical elementary divisors

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 253-267

The problem of reducing polynomial matrices to the canonical form by using semiscalar equivalent transformations is studied. A class of polynomial matrices is singled out, for which the canonical form with respect to semiscalar equivalence is indicated. This form enables one to solve the classification problem for collections of matrices over a field up to similarity.

### Decomposability of matrix polynomials with commuting coefficients into a product of linear factors

Ukr. Mat. Zh. - 2010. - 62, № 8. - pp. 1114–1123

The list of known sets of factorizable matrix polynomials is supplemented by new sets of polynomials of this sort. The known set of nonfactorizable matrix polynomials is extended. These results can be applied to the study of polynomial equations and systems of differential equations with constant coefficients.

### On Semiscalar and Quasidiagonal Equivalences of Matrices

Ukr. Mat. Zh. - 2000. - 52, № 10. - pp. 1435-1440

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.

### Decomposition of polynomial matrices into a direct sum of triangular summands

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1144–1148

By using the transformations*SA(x)R(x)*, where*S* and*R(x)* are invertible matrices, we reduce a polynomial matrix*A(x)* whose elementary divisors are pairwise relatively prime to a direct sum of irreducible triangular summands with invariant factors on the principal diagonals.

### Selection of regular direct summands from matrix polynomials

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 927–934

We study the problem of reducibility of matrix polynomials to a quasidiagonal form with regular diagonal blocks by a similarity transformation. The results obtained are applied to solving matrix algebraic equations of Riccati type.

### Decomposability of matrix polynomials with coefficients possessing quasi-simple structure

Ukr. Mat. Zh. - 1992. - 44, № 12. - pp. 1716–1718