- What can you say about \(n \times n\) matrices \(A\) that satisfy a given polynomial \(P(x) = 0\) over an algebraically closed field \(k = \bar{k}\)? How many similarity classes and how many such matrices are there?
- What about over a finite field \(\mathbb{F}_q\)? How many similarity classes and how many individual matrices satisfy \(P(A) = 0\)?
Solution
For any field \(K\), \[ P(A)=0\iff \mu_A\mid P, \] where \(\mu_A\) is the minimal polynomial of \(A\).
Over an algebraically closed field, write \[ P(x)=c\prod_{i=1}^r(x-\lambda_i)^{e_i}. \] Then a matrix \(A\in M_n(K)\) satisfies \(P(A)=0\) exactly when every Jordan block with eigenvalue \(\lambda_i\) has size at most \(e_i\), and no other eigenvalue occurs. Hence the similarity classes are parametrized by tuples of partitions \[ \nu_i\vdash n_i,\qquad \sum_i n_i=n, \] with largest part of \(\nu_i\) at most \(e_i\). In particular, there are finitely many similarity classes.
The individual matrices form the finite union of their conjugacy orbits \[ \mathcal O_A=\{SAS^{-1}:S\in\operatorname{GL}_n(K)\}. \] Thus there is no universal answer “infinitely many”: if all admissible matrices are scalar, the set may be finite (for example, \(P=x-a\) gives only \(aI\)). If a non-scalar admissible similarity class occurs and \(K\) is infinite, that orbit is positive-dimensional and has \(|K|\) points; over \(K=\mathbb C\) it therefore has continuum cardinality.
Over \(\mathbb F_q\), factor \[ P(x)=\prod_j f_j(x)^{e_j} \] into distinct monic irreducibles. Similarity classes are described by the corresponding rational-canonical/primary data: for each \(f_j\), choose a partition whose largest part is at most \(e_j\), subject to total dimension \[ \sum_j (\deg f_j)|\nu_j|=n. \] Again only finitely many classes occur. The exact number of individual matrices is \[ \sum_{[A]}\frac{|\operatorname{GL}_n(\mathbb F_q)|} {|C_{\operatorname{GL}_n(\mathbb F_q)}(A)|}, \] where the sum runs over the admissible similarity classes. This is a finite computable integer.