# Prokip V. M.

### On the Solvability of a System of Linear Equations Over the Domain Of Principal Ideals

Ukr. Mat. Zh. - 2014. - 66, № 4. - pp. 566–570

We propose new necessary and sufficient conditions for the solvability of a system of linear equations over the domain of principal ideals and an algorithm for the solution of this system.

### Diagonalizability of matrices over a principal ideal domain

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 283-288

A square matrix is said to be diagonalizable if it is similar to a diagonal matrix. We establish necessary and sufficient conditions for the diagonalizability of matrices over a principal ideal domain.

### Canonical form with respect to semiscalar equivalence for a matrix pencil with nonsingular first matrix

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1147-1152

Polynomial matrices $A(x)$ and $B(x)$ of size $n \times n$ over a field $\mathbb{F}$ are called semiscalar equivalent if there exist a nonsingular $n \times n$ matrix $P$ over $\mathbb{F}$ and an invertible $n \times n$ matrix $Q(x)$ over $\mathbb{F}[x]$ such that $A(x) = PB(x)Q(x)$. We give a canonical form with respect to the semiscalar equivalence for a matrix pencil $A(x) = A_0x - A_1$, where $A_0$ and $A_1$ are $n \times n$ matrices over $\mathbb{F}$, and $A_0$ is nonsingular.

### On One Class of Divisors of Polynomial Matrices over Integral Domains

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1099-1106

We establish conditions for the existence of a unital divisor for a polynomial matrix over an integral domain of characteristic zero in the case where its eigenvalues are known.

### Structure of Matrices and Their Divisors over the Domain of Principal Ideals

Ukr. Mat. Zh. - 2002. - 54, № 8. - pp. 1143-1148

We investigate the structure of matrices and their divisors over the domain of principal ideals.

### On Multiplicativity of Canonical Diagonal Forms of Matrices over the Domain of Principal Ideals. II

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 274-277

We investigate the structure of matrices over the domain of principal ideals that possess the property of multiplicativity of canonical diagonal forms.

### Polynomial matrices over a factorial domain and their factorization with given characteristic polynomials

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1438–1440

We establish conditions for the existence of a unital divisor with given characteristic polynomial of a polynomial matrix over a factorial domain.

### On the factorization of polynomial matrices over the domain of principal ideals

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1435-1439

We consider the problem of decomposition of polynomial matrices over the domain of principal ideals into a product of factors of lower degrees with given characteristic polynomials. We establish necessary and, under certain restrictions, sufficient conditions for the existence of the required factorization.

### To the problem of multiplicativity of canonical diagonal forms of matrices over the domain of principal ideals

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1581–1584

We study the structure of nonsingular matrices over the domain of principal ideals that possess the property of multiplicativity of canonical diagonal forms. In particular, we establish necessary and sufficient conditions of multiplicativity of canonical diagonal forms of nonsingular matrices over this domain.

### On the solvability of the Riccati matrix algebraic equation

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1591–1593

We obtain conditions of solvability of the Riccati matrix algebraic equation.

### A method for finding a common linear divisor of the matrix polynomials over an arbitrary field

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1181–1183

Conditions are found for existence of a common linear unital divisor with a given characteristic polynomial of regular matrix polynomials over an arbitrary field. The result obtained is also applied to finding a common solution of matrix polynomial equations.

### On the uniqueness of the unital divisor of a matrix polynomial over an arbitrary field

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 803–808

Conditions are established under which the unital divisor extracted from a matrix polynomial over an arbitrary field is determined uniquely by its characteristic polynomial. The result obtained is applied to the problem of solving matrix polynomial equations.

### Factorization of polynomial matrices over arbitrary fields

Petrichkovich V. M., Prokip V. M.

Ukr. Mat. Zh. - 1986. - 38, № 4. - pp. 478–483