Exercise 12, § 5

a) Let A and B be two square matrices of order n over a commutative field K, and let $f_i$ and $g_i$ ($1 \leq i \leq n$) be their respective similarity invariants. Show that the vector space (over K) of square matrices $U$, of order $n$, such that $UA = BU$, is isomorphic to the space $\overline{M} = M/N$ defined as follows : M is the space of square matrices $(u_{ij}(X))$ of order n over $K[X]$ such that $u_{ij}(X) f_j(X)$ is a multiple of $g_i(X)$ for each pair $i, j$ of indices ; and N is the subspace of M consisting of those matrices for which $u_{ij}(X)$ is a multiple of $g_j(X)$ for each pair $i, j$ of indices. (Consider $U, A$ and B as the matrices of endomorphisms $u, v$ and w respectively of $E = K^n$, and notice that u is a $K[X]$-linear map from $E_v$ into $E_w$; now express $E_v$ and $E_w$ as quotient modules.) b) When $B = A$, the matrices $U$ such that $UA = AU$ form a ring P ; then M is a ring of square matrices, and N a 2-sided ideal of M, such that P is isomorphic to the quotient ring $M/N$. Show that P is identical to the ring $K[A]$ generated by I, and A if and only if the minimal polynomial of A is equal to its characteristic polynomial. c) For arbitrary A and B, show that the dimension of $M_0$ is equal to $\sum_{i,j} m_{ij}$, where $m_{ij}$ is the degree of the gcd of $f_i$ and $g_j$ (same method as a)). d) Show that the greatest of the ranks of the matrices $U \in M_0$ is equal to $\sum_k d_k$, where $d_k$ is the degree of the gcd of $f_k$ and $g_k$ (use Ex. 8). e) Generalise the results of a), c) and d) to the case where A is a square matrix of order $n$, B is a square matrix of order m, and $U$ is a matrix with m rows and n columns.