The Direct Product of the Ring of m x m and n x n Matrices

# The Direct Product of the Ring of m x m and n x n Matrices

Recall from The Direct Product of Two Rings page that if $(R, +_1, *_1)$ and $(S, +_2, *_2)$ are both rings, then the direct product of these rings is the ring $(T, +, *)$ where:

(1)
\begin{align} \quad T = R \times S = \{ (a, b) : a \in R, b \in S \}. \end{align}

The operation of $+$ is defined for $(a, b), (c, d) \in T$ as:

(2)
\begin{align} \quad (a, b) + (c, d) = (a +_1 c, b +_2 d). \end{align}

The operation of $*$ is defined for $(a, b), (c, d) \in T$ as:

(3)
\begin{align} \quad (a, b) * (c, d) = (a *_1 c, b *_2 d). \end{align}

We will now look at a specific example of the direct product of two rings which has a very nice interpretation. Let $M_{mm}$ be the set of all $m \times m$ matrices with real coefficients and let $M_{nn}$ be the set of all $n \times n$ matrices with real coefficients. On The Ring of n x n Matrices page, we have already noted that these sets form a ring if we define the operations $+_1$, $+_2$, $*_1$ and $*_2$ to be standard matrix addition and standard matrix multiplication, i.e., $(M_{mm}, +_1, *_1)$ and $(M_{nn}, +_2, *_2)$. Now let $T$ be the direct product of these rings:

(4)
\begin{align} \quad T = M_{mm} \times M_{nn} = \{ (A, B) : A \in M_{mm}, B \in M_{nn} \} \end{align}

We already have a general idea of what the rings $(M_{mm}, +_1, *_1)$ and $(M_{nn}, +_2, *_2)$ look like, but, what does $(T, +, *)$ look like? One way to visualize $T$ is by representing it as the set of all $m + n \times m + n$ block matrices of the following form:

(5)
\begin{align} \quad (A, B) = \begin{bmatrix} A_{m \times m} & 0_{m \times n}\\ 0_{n \times m} & B_{n \times n} \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1m} & 0 & 0 & \cdots & 0\\ a_{21} & a_{22} & \cdots & a_{2m} & 0 & 0 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mm} & 0 & 0 & \cdots & 0\\ 0 & 0 & \cdots & 0 & b_{11} & b_{12} & \cdots & b_{1n}\\ 0 & 0 & \cdots & 0 & b_{21} & b_{22} &\cdots & b_{2n}\\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 & b_{n1 }& b_{n2} & \cdots & b_{nn} \end{bmatrix} \end{align}

In we consider elements in $T$ to be of the form above, then all of the ring axioms for $T$ have a nice representation. For example, the identity of $+$ in $T$ is the element $(0_{m\times m}, 0_{n \times n}$ which would translate above into the $m + n \times m + n$ zero matrix:

(6)
\begin{align} \quad (0_{m \times m}, 0_{n \times n}) = \begin{bmatrix} A_{m \times m} & 0_{m \times n}\\ 0_{n \times m} & B_{n \times n} \end{bmatrix} = \begin{bmatrix} 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0\\ 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0\\ 0 & 0 & \cdots & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \cdots & 0 & 0 & 0 &\cdots & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 & 0 & 0 & \cdots 0 \end{bmatrix}_{(m+n) \times (m+n)} \end{align}

Similarly, the identity of $*$ in $T$ is the element $(I_{m \times m}, I_{n \times n})$ which would translate above into the $m + n \times m + n$ identity matrix whose main diagonal consists of $1$s and whose other entries consist of $0$s.

(7)
\begin{align} \quad (I_{m\times m}, I_{n\times n}) = \begin{bmatrix} A_{m \times m} & 0_{m \times n}\\ 0_{n \times m} & B_{n \times n} \end{bmatrix} = \begin{bmatrix} 1 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0\\ 0 & 1 & \cdots & 0 & 0 & 0 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 & 0 & 0 & \cdots & 0\\ 0 & 0 & \cdots & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & \cdots & 0 & 0 & 1 &\cdots & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 & 0 & 0 & \cdots 1 \end{bmatrix}_{(m+n) \times (m+n)} \end{align}

If we use the representation for $T$ above then $+$ and $*$ defined on $T$ correspond to the $m \times m$ standard matrix addition and matrix multiplication of matrices $A \in M_{mm}$ and also to the $n \times n$ standard matrix addition and matrix multiplication of matrices $B \in M_{nn}$. In fact, all of the ring axioms on $T$ nicely correspond to the ring axioms on $R$ and $S$ as you should verify.