direct sum of matrices
Direct sum of matrices
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
$$\left(\begin{array}{cc}\hfill A\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill B\hfill \end{array}\right)$$ 
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 $A=\left(\begin{array}{cc}\hfill 3\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 5\hfill \end{array}\right)$ and $B=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 4\hfill & \hfill 0\hfill \\ \hfill 7\hfill & \hfill 8\hfill \end{array}\right)$, then
$$\left(\begin{array}{cc}\hfill A\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill B\hfill \end{array}\right)=\left(\begin{array}{cccc}\hfill 3\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 7\hfill & \hfill 8\hfill \end{array}\right)$$ 
Remark. It is not hard to see that the $\oplus $ operation on matrices is associative:
$$(A\oplus B)\oplus C=A\oplus (B\oplus C),$$ 
because both sides lead to
$$\left(\begin{array}{ccc}\hfill A\hfill & \hfill O\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill B\hfill & \hfill O\hfill \\ \hfill O\hfill & \hfill O\hfill & \hfill C\hfill \end{array}\right)$$ 
In fact, we can inductively define the direct sum of $n$ matrices unambiguously.
Direct sums of linear transformations
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
$${T}_{A\oplus B}:{k}^{n+q}\to {k}^{m+p}$$ 
using the standard ordered basis, such that

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

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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 $v\in {V}_{1}\oplus {V}_{2}$,
$$({T}_{1}\oplus {T}_{2})({v}_{1},{v}_{2}):=({T}_{1}({v}_{1}),{T}_{2}({v}_{2}))$$ 
where ${v}_{i}\in {V}_{i}$. Based on this definition, it is not hard to see that
$${T}_{A\oplus B}={T}_{A}\oplus {T}_{B}$$ 
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 $$ is an ordered basis for ${V}_{1}\oplus {V}_{2}$, and
$${[{T}_{1}\oplus {T}_{2}]}_{\beta}={[{T}_{1}]}_{{\beta}_{1}}\oplus {[{T}_{2}]}_{{\beta}_{2}}.$$ 
Title  direct sum of matrices 

Canonical name  DirectSumOfMatrices 
Date of creation  20130322 17:36:48 
Last modified on  20130322 17:36:48 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 1501 
Related topic  DirectSum 