An explicit expression for the minimal polynomial of the Kronecker product of matrices. Explicit formulas for matrix logarithm and matrix exponential

Abstract

Using $\mathcal{P}$-canonical forms of matrices, we derive the minimal polynomial of the Kronecker product of a given family of matrices in terms of the minimal polynomials of these matrices. This, allows us to prove that the product $\prod _{i=1}^{m}L(P_i)$, $L(P_i)$ is the set of linear recurrence sequences over a field $F$ with characteristic polynomial $P_i$, is equal to $L(P)$ where $P$ is the minimal polynomial of the Kronecker product of the companion matrices of $P_i$, $1\le i\le m$. Also, we show how we deduce from the $\mathcal{P}$-canonical form of an arbitrary complex matrix $A$, the $\mathcal{P}$-canonical form of the matrix function $e^{tA}$ and a logarithm of $A$.

19 pages; references added; results added; examples added

References in corpus (1)

• Some algebraic results concerning linear recurrence sequences