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

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