Separating a unital divisor from a rectangular polynomial matrix

It is required to separate a unital divisor with a preassigned characteristic polynomial from a rectangular polynomial matrix over a field. Necessary and sufficient conditions of existence, under certain restrictions, are obtained for such a divisor, as well as a method for constructing it. By the approach used in this paper it is possible to completely solve this problem for rectangular polynomial matrices, all of whose elementary divisiors are pair-wise relatively prime. The results obtained are illustrated by solving matrix equations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1089–1094, August, 1990.     

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

Conditions are found which guarantee the existence of a common linear unital divisor with a given characteristic polynomial for the regular matrix polynomials over an arbitrary field. The result obtained is applied when finding a common solution of matrix polynomial equations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1181–1183, August, 1993.     

Parallel factorizations of matrix polynomials over an arbitrary field

We study the problem of the decomposition of a matrix polynomial over an arbitrary field into a product of factors of lower degrees with preassigned characteristic polynomials. We find necessary conditions for the existence of the required factorization, which are also sufficient for certain classes of matrix polynomials. The proposed method makes it possible to solve the problem completely for matrix polynomials with one nonconstant invariant factor. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

Generalized polynomial Bezoutian with respect to a Jacobson chain basis over an arbitrary field

We introduce a so-called generalized polynomial Bezoutian with respect to a Jacobson chain basis over an arbitrary field. Some characterization of this kind of matrix, such as the Barnett-type factorization and the intertwining relation with generalized hypercompanion matrix, are obtained. The diagonal reduction formula via the generalized confluent Vandermonde matrix similar to that of classical Bezoutian is presented. The method used is based on polynomial module and operator representation.     

On common unital divisors of polynomial matrices

We establish a condition for the existence of common unital divisors of polynomial matrices over an arbitrary field, with the divisors having a prescribed characteristic polynomial. The results obtained are applied to find a common solution of matrix polynomial equations.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 20–24.     

A fast parallel algorithm to compute the rank of a matrix over an arbitrary field

It is shown that the rank of a matrix over an arbitrary field can be computed inO(log² n) time using a polynomial number of processors. Also appeared in ACM Symposium on Theory of Computing, May 28–30, 1986 Berkeley, California. Research supported by Miller Fellowship, University of California, Berkeley.     

A note on polynomial matrix functions over a finite field

Let F_n denote the ring of n×n matrices over the finite field F=GF(q) and let A(x)=A_Nx^N+ ?+ A₁x+A₀?F_n[x]. A function $\text{?:F}{}_{\mathrm{n}}\text{→F}{}_{\mathrm{n}}$ is called a right polynomial function iff there exists an A(x)?F_n[x] such that $\text{?(B)=A}{}_{\mathrm{N}}\text{B}{}^{\mathrm{N}}\text{+?+A}{}_1\text{B+ A}{}_0$ for every B?F_n. This paper obtains unique representations for and determines the number of right polynomial functions.     

Characterization of commutative matrix subalgebras of maximal length over an arbitrary field

Commutative subalgebras of length n − 1 in the full matrix algebra of order n over an arbitrary field are characterized in terms of generating elements.     

The eigenstructure of an arbitrary polynomial matrix: computational aspects

We give a new numerical method to compute the eigenstructure—i.e. the zero structure, the polar structure, and the left and right null space structure—of a polynomial matrix P(λ). These structural elements are of fundamental importance in several systems and control problems involving polynomial matrices. The approach is more general than previous numerical methods because it can be applied to an arbitrary m × n polynomial matrix P(λ) with normal rank r smaller than m and/or n. The algorithm is then shown to compute the structure of the left and right null spaces of P(λ) as well. The speed and accuracy of this new approach are also discussed.     

On the matrix ring over a finite field

Let F_n be the ring of n × n matrices over the finite field F; let o(F_n) be the number of elements in F_n, and s(F_n) be the number of singular matrices in F_n. We prove that $\text{o(F}{}_{\mathrm{n}}\text{)<s(F}{}_{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1+1</mtext><mtext>n(n-1)</mtext>}}$ if n ? 2, and if n = 2 and o(F) ? 3, then .     

On linear combinations of two idempotent matrices over an arbitrary field

Given an arbitrary field K and non-zero scalars α and β, we give necessary and sufficient conditions for a matrix A∈M_n(K) to be a linear combination of two idempotents with coefficients α and β. This extends results previously obtained by Hartwig and Putcha in two ways: the field K considered here is arbitrary (possibly of characteristic 2), and the case α≠±β is taken into account.     

On computation of the greatest common divisor of several polynomials over a finite field

We propose a probabilistic algorithm to reduce computing the greatest common divisor of m polynomials over a finite field (which requires computing m−1 pairwise greatest common divisors) to computing the greatest common divisor of two polynomials over the same field.     

On the Toeplitz embedding of an arbitrary matrix

It has been shown by Delosme and Morf that an arbitrary block matrix can be embedded into a block Toeplitz matrix; the dimension of this embedding depends on the complexity of the matrix structure compared to the block Toeplitz structure. Due to the special form of the embedding matrix, the algebra of matrix polynomials relative to block Toeplitz matrices can be interpreted directly in terms of the original matrix and therefore can be extended to arbitrary matrices. In fact, these polynomials turn out to provide an appropriate framework for the recently proposed generalized Levinson algorithm solving the general matrix inversion problem.     

On the hermite matrix of a polynomial

We investigate the location of the eigenvalues of the Hermite matrix of a given complex polynomial, the question under what conditions a given polynomial and the characteristic polynomial of its Hermite matrix are identical, and the question under what conditions the Hermite matrix has only one distinct eigenvalue.     

Exponential sum estimates over a subgroup in an arbitrary finite field

Let F _q be the finite field consisting of q = p ^r elements and yy an additive character of the field F _q . Take an arbitrary multiplicative subgroup H of size |H| > q ^{C/(log log q)} for some constant C > 0 not largely contained in any multiplicative shift of a subfield. We show that |Σ _h∈H yy(h)| = o(|H|). This means that H is equidistributed in F _q .