Let and be square matrices over a ring . Recall that the trace of a square matrix is the sum of its diagonal entries. Let denote the Kronecker product of and . Prove that .
First we will give a concrete recursive characterization of the Kronecker product.
Let and be matrices. If , then . If , then .
Now to the main result; we will proceed by induction on the dimensions of . If , then certainly , and .
For the inductive step, suppose the result holds when has dimension , and let be a matrix with dimension . Say . Then as desired.