In this note, we prove that a square matrix of size n over a field containing at least 2n elements can be expressed as the product of two matrices similar to companion matrices, that is to say matrices with the same minimal and characteristic polynomial, if and only if the rank of A is greater than n-2, using only elementary facts. We will also give some partial results valid over smaller fields.
Flavien Mabilat (Thu,) studied this question.
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