On the similarity of matrices AB and BA over a field
Journal Title: Карпатські математичні публікації - Year 2018, Vol 10, Issue 2
Abstract
Let A and B be n-by-n matrices over a field. The study of the relationship between the products of matrices AB and BA has a long history. It is well-known that AB and BA have equal characteristic polynomials (and, therefore, eigenvalues, traces, etc.). One beautiful result was obtained by H. Flanders in 1951. He determined the relationship between the elementary divisors of AB and BA, which can be seen as a criterion when two matrices C and D can be realized as C=AB and D=BA. If one of the matrices (A or B) is invertible, then the matrices AB and BA are similar. If both A and B are singular then matrices AB and BA are not always similar. We give conditions under which matrices AB and BA are similar. The rank of matrices plays an important role in this investigation.
Authors and Affiliations
V. M. Prokip
Keywords
Related Articles
- EP ID EP535496
- DOI 10.15330/cmp.10.2.352-359
- Views 28
- Downloads 0
Wiman's inequality for analytic functions in D×C with rapidly oscillating coefficients
Let A2 be a class of analytic function f represented by power series of the from f(z)=f(z1,z2)=+∞∑n+m=0anmzn1zm2 with the domain of convergence T={z∈C2:|z1|<1,|z2|<+∞} such that ∂∂z2f(z1,z2)≢0 in T and there exists r0...
The nonlocal boundary problem with perturbations of antiperiodicity conditions for the eliptic equation with constant coefficients
In this article, we investigate a problem with nonlocal boundary conditions which are perturbations of antiperiodical conditions in bounded m-dimensional parallelepiped using Fourier method. We describe properties of a t...
On some perturbations of a stable process and solutions to the Cauchy problem for a class of pseudo-differential equations
A fundamental solution for some class of pseudo-differential equations is constructed by the method based on the theory of perturbations. We consider a symmetric α-stable process in multidimensional Euclidean space. Its...
Homomorphisms and functional calculus in algebras of entire functions on Banach spaces
In the paper the homomorphisms of algebras of entire functions on Banach spaces to a commutative Banach algebra are studied. In particular, it is proposed a method of constructing of homomorphisms vanishing on homogeneou...
Parabolic by Shilov systems with variable coefficients
Because of the parabolic instability of the Shilov systems to change their coefficients, the definition parabolicity of Shilov for systems with time-dependent $t$ coefficients, unlike the definition parabolicity of Petro...