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Homedirect sum of matrices
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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
$\begin{pmatrix}A&O\\ O&B\end{pmatrix}$ 
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
$\begin{pmatrix}A&O\\ O&B\end{pmatrix}=\begin{pmatrix}3&1&0&0\\ 2&5&0&0\\ 0&0&1&2\\ 0&0&4&2\\ 0&0&7&8\\ \end{pmatrix}$ 
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
$\begin{pmatrix}A&O&O\\ O&B&O\\ O&O&C\end{pmatrix}$ 
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

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 $v\in V_{1}\oplus V_{2}$,
$(T_{1}\oplus T_{2})(v_{1},v_{2}):=(T_{1}(v_{1}),T_{2}(v_{2}))$ 
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 $v_{i}<v_{j}$ is an ordered basis for $V_{1}\oplus V_{2}$, and
$[T_{1}\oplus T_{2}]_{{\beta}}=[T_{1}]_{{\beta_{1}}}\oplus[T_{2}]_{{\beta_{2}}}.$ 
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Comments
Adding to Your Entry
Hello 
I am wondering if you would mind my adding a short exposition to you post concerning Sum and Direct Sum.
I have proved some fundamental results and would like to post them and get some input.
Patrick J. O'Hara
Adding to Your Entry
Dear Chi Woo:
Thank you so much. I am working on two(2) proofs related to linear operators and matrices that appear directly related to what your addition is stating.
The challenge will then be to articulate this connection.
It may be some time before I actually make the addition. If you like, I can notify you beforehand.
PJO
Contributing a paper
Wd like to introduce self as a nonacademic mathematician
(site:www.crorepatibaniye.com/failurefunctions)
Q: I have a paper ready for publishing on arxiv.I would also like to publish it on in planetmath ( preferably only pdf).
a) How do I go about it? b)Is it ethically ok?
Devaraj
Re: Contributing a paper
a. Click on "add to papers" on the menu bar on the left and
fill in the form that appears.
b. As far as PlanetMath is concerned, all that matters is that
it is legal to distribute your article online. Basically, if
you haven't signed a contract with a publisher (which usually
gives the publisher the copyright) and provide a rights statement
which gives permission to distribute your work electronically,
there should be no problem.
Re: Contributing a paper
Devaraj
Yes if you are a registered participate and agree to the user terms that are part of the membership agreement. So you would need to register first.
Then, if it is an actual paper, there is a section "add to" on the side. Just click "Papers" and follow directions. The file upload is on the very bottom of the "add to papers" page.
If you have further questions, go to the "Forum" section under "Help using planetmath" and an Administrator can help you.
Best wishes.
PJO
Re: Contributing a paper
What are the steps for registration pl? Tks,
Devaraj
Re: Contributing a paper
I have not yet received guidance on
becoming a member pl.
A.K.Devaraj
Re: Contributing a paper
If you are able to post messages in this forum, you are a member of PlanetMath.
If you want to contribute a paper or an exposition here, click on the appropriate links on the left pane and proceed. The same follows if you want to contribute a Encyclopedia entry, except in this case you'd need to know some LaTeX. There's another way to add entries: if you want to write something that is related to an existing entry, scroll to the bottom of that entry, and click add (or add example), and then proceed as usual.
Let us know if you have any other questions.
Re: Contributing a paper
Thank you very much;I will try as directed.
A.K.Devaraj