Products of matrices with prescribed spectra and ranks

This paper studies the possibility of writing a given square matrix as the product of two matrices with prescribed spectra and ranks. It extends some previously known results.     

Factorization of singular integer matrices

It is well known that a singular integer matrix can be factorized into a product of integer idempotent matrices. In this paper, we prove that every n  × n (n > 2) singular integer matrix can be written as a product of 3n + 1 integer idempotent matrices. This theorem has some application in the field of synthesizing VLSI arrays and systolic arrays.     

Factorization of a rational matrix: The singular case

We study the problem of minimal factorization of an arbitrary rational matrix R(), i. e. where R() is not necessarily square or invertible. Following the definition of minimality used here, we show that the problem can be solved via a generalized eigenvalue problem which will be singular when R() is singular. The concept of invariant subspace, which has been used in the solution of the minimal factorization problem for regular matrices, is now replaced by a reducing subspace, a recently introduced concept which is a logical extension of invariant and deflating subspaces to the singular pencil case.     

Extreme determinants on the convex hull of matrices with prescribed singular values

In a recent paper[1] the author carried through a comprehesive analysis of the diagonal elements of matrices having prescribed singular values, and as an application of this analysis a characterization was obtained of matrices with prescribed singular values. In the present note we obtain the greatest and least values for the determinants of the matrices in such convex hulls.     

Extreme determinants on the convex hull of matrices with prescribed singular values

In a recent paper[1] the author carried through a comprehesive analysis of the diagonal elements of matrices having prescribed singular values, and as an application of this analysis a characterization was obtained of matrices with prescribed singular values. In the present note we obtain the greatest and least values for the determinants of the matrices in such convex hulls.     

Expansion of weighted pseudoinverse matrices with singular weights into matrix power products and iteration methods

We obtain expansions of weighted pseudoinverse matrices with singular weights into matrix power products with negative exponents and arbitrary positive parameters. We show that the rate of convergence of these expansions depends on a parameter. On the basis of the proposed expansions, we construct and investigate iteration methods with quadratic rate of convergence for the calculation of weighted pseudoinverse matrices and weighted normal pseudosolutions. Iteration methods for the calculation of weighted normal pseudosolutions are adapted to the solution of least-squares problems with constraints. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 9, pp. 1269–1289, September, 2007.     

Graphs with nilpotent adjacency matrices

Given any integer t ≥ 2 and any prime number p, a graph Γ_p,t is constructed whose adjacency matrix is nilpotent of index t over Z_p' the field of p elements.     

Decomposition of polynomial matrices into factors with prescribed canonical diagonal forms

We give a test for decomposability of a polynomial matrix over an arbitrary infinite field into factors with prescribed canonical diagonal forms whose product is the canonical diagonal form of the given matrix. We exhibit a method of actually constructing such decompositions of polynomial matrices.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 17–19.     

Products of nilpotent matrices

We show that any complex singular square matrix T is a product of two nilpotent matrices A and B with rank A = rank B = rank T except when T is a 2×2 nilpotent matrix of rank one.     

Sums of nilpotent matrices

We study matrices over general rings which are sums of nilpotent matrices. We show that over commutative rings all matrices with nilpotent trace are sums of three nilpotent matrices. We characterize 2-by-2 matrices with integer entries which are sums of two nilpotents via the solvability of a quadratic Diophantine equation. Some exemples in the case of matrices over noncommutative rings are given.     

Commutators of nilpotent matrices

A matrix of zero trace is, with certain exceptions, the commutator of nilpotent matrices. It may (so far as we know) be necessary to go outside the original field for the entries of the nilpotent matrices.     

Factorization of block matrices

An LU-type factorization theorem due to Elsner and to Gohberg and Goldberg is generalized to block matrices. One form of the general factorization takes the form LMU, where L is block lower-triangular, U is block upper-triangular, and M is a subpermutation matrix each of whose blocks is diagonal. A factorization is also given where the middle term is a block diagonal subpermutation matrix, and the factorization is applied to Wiener-Hopf equations. The nonuniqueness of the middle term in the factorization is analyzed. A special factorization for self-adjoint block matrices is also obtained.     

Existence of matrices with prescribed entries

It is proved that, apart from for some exceptional cases, there always exists an n×n nonderogatory matrix over an arbitrary field with n prescribed entries and prescribed characteristic polynomial.     

On a matrix nilpotent filter

We define two families of homogeneous ideals of the algebra of polynomials generated by power entries of the general matrix and its operator invariants. We study the combinatorial characteristics of these ideals and, in greater detail, the case of second order.     

Factorization of matrices of quaternions

We review known factorization results for quaternion matrices. Specifically, we derive the Jordan canonical form, polar decomposition, singular value decomposition, and QR factorization. We prove that there is a Schur factorization for commuting matrices, and from this derive the spectral theorem. We do not consider algorithms, but do point to some of the numerical literature.