Let $A$ be an $m\times n$ matrix and $B$ be a $p\times q$ matrix. By the direct sum of $A$ and $B$, written $A\oplus B$, we mean the $(m+p)\times(n+q)$ matrix of the form

where the $O$’s represent zero matrices. The $O$ on the top right is an $m\times q$ matrix, while the $O$ on the bottom left is $n\times p$.

For example, if $A=\begin{pmatrix}3&-1\\ 2&5\end{pmatrix}$ and $B=\begin{pmatrix}1&2\\ 4&0\\ -7&8\end{pmatrix}$, then

Remark. It is not hard to see that the $\oplus$ operation on matrices is associative:

because both sides lead to

In fact, we can inductively define the direct sum of $n$ matrices unambiguously.

The direct sum of matrices is closely related to the direct sum of vector spaces and linear transformations. Let $A$ and $B$ be as above, over some field $k$. We may view $A$ and $B$ as linear transformations $T_{A}:k^{n}\to k^{m}$ and $T_{B}:k^{q}\to k^{p}$ using the standard ordered bases. Then $A\oplus B$ may be viewed as the linear transformation

using the standard ordered basis, such that

• the restriction of $T_{{A\oplus B}}$ to the subspace $k^{n}$ (embedded in $k^{{n+q}}$) is $T_{A}$, and

• the restriction of $T_{{A\oplus B}}$ to $k^{q}$ is $T_{B}$.

The above suggests that we can define direct sums on linear transformations. Let $T_{1}:V_{1}\to W_{1}$ and $T_{2}:V_{2}\to W_{2}$ be linear transformations, where $V_{i}$ and $W_{j}$ are finite dimensional vector spaces over some field $k$ such that $V_{1}\cap V_{2}=0$. Then define $T_{1}\oplus T_{2}:V_{1}\oplus V_{2}\to W_{1}\oplus W_{2}$ such that for any $v\in V_{1}\oplus V_{2}$,

where $v_{i}\in V_{i}$. Based on this definition, it is not hard to see that

for any matrices $A$ and $B$.

More generally, if $\beta_{i}$ is an ordered basis for $V_{i}$, then $\beta:=\beta_{1}\cup\beta_{2}$ extending the linear orders on $\beta_{i}$, such that if $v_{i}\in\beta_{1}$ and $v_{j}\in\beta_{2}$, then $v_{i}<v_{j}$ is an ordered basis for $V_{1}\oplus V_{2}$, and