How to construct a class of matrices satisfying a given matrix equation.

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The given matrix equation implies that the minimal polynomial of $A$ divides $f(x):=p^2 x^{p^2} - q^2 x^{q^2} - r^2$. It follows that $A$ can be constructed as a block diagonal matrix $\begin{bmatrix} C_g & 0\\ 0 & I_{n-d} \end{bmatrix}$ for any monic divisor $g(x)\mid f(x)$ of degree $d:=\deg g(x)\leq n$, where $C_g$ is the companion matrix of $g(x)$.

Here is a sample code that constructs and prints such matrices:

def compA(n,p,q,r):
    assert p^2 == q^2 + r^2
    R.<x> = PolynomialRing(ZZ)
    f = p^2*x^(p^2) - q^2*x^(q^2) - r^2
    for g in divisors(f):
        if g.degree()>n or not g.is_monic():
            continue
        A = block_diagonal_matrix(companion_matrix(g), identity_matrix(n-g.degree()))
        print(A)

More generally, $A$ can be taken as a block diagonal matrix with blocks $C_{g_1}, \dots, C_{g_k}$, where each $g_i(x)$ is a monic divisor of $f(x)$, and $\sum_{i=1}^k \deg g_i(x) = n$. Matrices similar to such $A$ will also satisfy the given matrix equation.