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Consider a set $T$ of square matrices over finite field $\mathbb{F_p}$. Clearly the cardinality of the set $T$ is $p^{n^2}$ where the square matrices are of size $n$.

Question is: How many non-similar ($A=P^{-1}.B.P$ for some $P$ $\in$ $T$) matrices we would have?

Next, suppose there are two matrices $A$ and $B$ such that their minimal polynomial is same $f$ (say). Question is: Are $A$ and $B$ similar matrices?

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To your second question: no, the minimal polynomial does not suffice. As an example, consider the following two matrices $$ A = \begin{bmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \end{bmatrix}, \qquad B = \begin{bmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ \end{bmatrix}. $$ They have the same minimal polynomial, namely $x^{2}$, but they are not similar. Similarity depends in general not on a single polynomial, but on a sequence of polynomials, each dividing the next one. For $A$ the sequence is $x, x, x^{2}$, for $B$ it is $x^{2}, x^{2}$.

A good reference is Jacobson's Basic Algebra I.


As to your first question, I refer you to this question and the answer provided there.

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  • $\begingroup$ Would you please mention which condition should be added to get A and B as similar? $\endgroup$
    – Fukuzita
    Jun 5, 2014 at 8:39
  • $\begingroup$ As I said in the answer, the two matrices are similar if and only if they share a certain sequence of monic polynomials, called invariant factors. Nothing difficult, but you should read the details in a book like the one I mentioned. $\endgroup$ Jun 5, 2014 at 8:43

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