Let $\vek{e}_i$ denote the (column) vector whose $i$ th position is $1$ and where all other positions are $0$ . Denote by $[n]$ the set . Denote by $\Mat_n(\mc{K})$ the set of all $n \times n$ matrices over $\mc{K}$ , and by $\GL_n(\mc{K})$ the set of all invertible elements of $\Mat_n(\mc{K})$ .

Theorem 1   Let $\mc{K}$ be a field, let be pairwise commuting matrices, and let $\mc{L}$ be a field extension of $\mc{K}$ in which the characteristic polynomials of all $A_k$ split. Then there exists an equivalence relation $\sim$ on $[n]$ and a matrix such that:

1. If and then .
2. If then .
3. If then and .

In other words there exists a simultaneous upper triangular block-diagonalisation of the matrices in which each block is characterised by the particular values of the diagonal elements.

The proof of this theorem is the obvious combination of the following two lemmas.

Lemma 2   Let $\mc{K}$ be a field, let be pairwise commuting matrices, and let $\mc{L}$ be a field extension of $\mc{K}$ in which the characteristic polynomials of all $A_k$ split. Then there exists some such that

1. is upper triangular for all , and
2. if are such that and for all , then for all as well.

Let for all and define \begin{equation*} i \sim j \quad\text{if and only if}\quad \Trans{\vek{e}_i} P^{-1} A_k P \vek{e}_i = \Trans{\vek{e}_j} P^{-1} A_k P \vek{e}_j \text{ for all }k \in [r]\text{.} \end{equation*}

Lemma 3   Let $\mc{L}$ be a field, let $n$ be a positive integer, and let $\sim$ be an equivalence relation on $[n]$ such that if and then . Let be pairwise commuting upper triangular matrices. If these matrices and $\sim$ are related such that \begin{equation*} i \sim j \quad\text{if and only if}\quad \Trans{\vek{e}_i} B_k \vek{e}_i = \Trans{\vek{e}_j} B_k \vek{e}_j \text{ for all }k \in [r]\text{,} \end{equation*}then there exists a matrix such that:

1. If then and .
2. If then .

The wanted $R$ is then $PQ$ .