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In [[mathematics]], '''matrix addition''' is the operation of adding two [[matrix (mathematics)|matrices]] by adding the corresponding entries together. However, there are other operations which could also be considered as a kind of [[addition]] for matrices, the [[direct sum]] and the [[Kronecker sum]].
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==Entrywise sum==
The usual matrix addition is defined for two matrices of the same dimensions. The sum of two ''m'' × ''n'' (pronounced "m by n") matrices '''A''' and '''B''', denoted by '''A''' + '''B''', is again an ''m'' × ''n'' matrix computed by adding corresponding elements:{{sfn |Lipschutz |Lipson}}<ref>{{cite book |title=Mathematical methods for physics and engineering |first1=K.F. |last1=Riley |first2=M.P.|last2=Hobson |first3=S.J. |last3=Bence | publisher=Cambridge University Press |year=2010 |isbn=978-0-521-86153-3}}</ref>
 
:<math>\begin{align}
\bold{A}+\bold{B} & = \begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn} \\
\end{bmatrix} +
 
\begin{bmatrix}
b_{11} & b_{12} & \cdots & b_{1n} \\
b_{21} & b_{22} & \cdots & b_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
b_{m1} & b_{m2} & \cdots & b_{mn} \\
\end{bmatrix} \\
& = \begin{bmatrix}
a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n} \\
a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots & a_{mn} + b_{mn} \\
\end{bmatrix} \\
 
\end{align}\,\!</math>
 
For example:
 
:<math>
  \begin{bmatrix}
    1 & 3 \\
    1 & 0 \\
    1 & 2
  \end{bmatrix}
+
  \begin{bmatrix}
    0 & 0 \\
    7 & 5 \\
    2 & 1
  \end{bmatrix}
=
  \begin{bmatrix}
    1+0 & 3+0 \\
    1+7 & 0+5 \\
    1+2 & 2+1
  \end{bmatrix}
=
  \begin{bmatrix}
    1 & 3 \\
    8 & 5 \\
    3 & 3
  \end{bmatrix}
</math>
 
We can also subtract one matrix from another, as long as they have the same dimensions. '''A''' &minus; '''B''' is computed by subtracting corresponding elements of '''A''' and '''B''', and has the same dimensions as '''A''' and '''B'''. For example:
 
:<math>
\begin{bmatrix}
1 & 3 \\
1 & 0 \\   
1 & 2
\end{bmatrix}
-
\begin{bmatrix}
0 & 0 \\
7 & 5 \\
2 & 1
\end{bmatrix}
=
\begin{bmatrix}
1-0 & 3-0 \\
1-7 & 0-5 \\
1-2 & 2-1
\end{bmatrix}
=
\begin{bmatrix}
1 & 3 \\
-6 & -5 \\
-1 & 1
\end{bmatrix}
</math>
 
==<span id="directsum" />Direct sum==
Another operation, which is used less often, is the direct sum (denoted by ⊕). Note the Kronecker sum is also denoted ⊕; the context should make the usage clear. The direct sum of any pair of matrices '''A''' of size ''m'' &times; ''n'' and '''B''' of size ''p'' &times; ''q'' is a matrix of size (''m'' + ''p'') &times; (''n'' + ''q'') defined as <ref>{{MathWorld |id=MatrixDirectSum |title=Matrix Direct Sum}}</ref>{{sfn |Lipschutz |Lipson}}
 
:<math>
  \bold{A} \oplus \bold{B} =
  \begin{bmatrix} \bold{A} & \boldsymbol{0} \\ \boldsymbol{0} & \bold{B} \end{bmatrix} =
  \begin{bmatrix}
    a_{11} & \cdots & a_{1n} &      0 & \cdots &      0 \\
    \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
    a_{m 1} & \cdots & a_{mn} &      0 & \cdots &      0 \\
          0 & \cdots &      0 & b_{11} & \cdots &  b_{1q} \\
    \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
          0 & \cdots &      0 & b_{p1} & \cdots &  b_{pq}
  \end{bmatrix}
</math>
 
For instance,
 
:<math>
  \begin{bmatrix}
    1 & 3 & 2 \\
    2 & 3 & 1
  \end{bmatrix}
\oplus
  \begin{bmatrix}
    1 & 6 \\
    0 & 1
  \end{bmatrix}
=
  \begin{bmatrix}
    1 & 3 & 2 & 0 & 0 \\
    2 & 3 & 1 & 0 & 0 \\
    0 & 0 & 0 & 1 & 6 \\
    0 & 0 & 0 & 0 & 1
  \end{bmatrix}
</math>
 
The direct sum of matrices is a special type of [[block matrix]], in particular the direct sum of square matrices is a [[Block matrix#Block diagonal matrices|block diagonal matrix]].
 
The [[adjacency matrix]] of the union of disjoint [[graph (mathematics)|graphs]] or [[multigraph]]s is the direct sum of their adjacency matrices. Any element in the [[Direct sum of modules|direct sum]] of two [[vector space]]s of matrices can be represented as a direct sum of two matrices.
 
In general, the direct sum of ''n'' matrices is:{{sfn |Lipschutz |Lipson}}
:<math>
\bigoplus_{i=1}^{n} \bold{A}_{i} = {\rm diag}( \bold{A}_1, \bold{A}_2, \bold{A}_3 \cdots \bold{A}_n)=
\begin{bmatrix}
\bold{A}_1 & \boldsymbol{0} & \cdots & \boldsymbol{0} \\
\boldsymbol{0} & \bold{A}_2 & \cdots & \boldsymbol{0} \\
\vdots & \vdots & \ddots & \vdots \\
\boldsymbol{0} & \boldsymbol{0} & \cdots & \bold{A}_n \\
\end{bmatrix}\,\!</math>
 
where the zeros are actually blocks of zeros, i.e. zero matricies.
 
NB: Sometimes in this context, boldtype for matrices is dropped, matricies are written in italic.
 
==Kronecker sum==
{{main|Kronecker sum}}
The Kronecker sum is different from the direct sum but is also denoted by ⊕. It is defined using the [[Kronecker product]] ⊗ and normal matrix addition. If '''A''' is ''n''-by-''n'', '''B''' is ''m''-by-''m'' and  <math>\mathbf{I}_k</math> denotes the ''k''-by-''k'' identity matrix then the Kronecker sum is defined by:
:<math> \mathbf{A} \oplus \mathbf{B} = \mathbf{A} \otimes \mathbf{I}_m + \mathbf{I}_n \otimes \mathbf{B}. </math>
 
==See also==
* [[Matrix multiplication]]
* [[Vector addition]]
 
==Notes==
{{reflist}}
 
==References==
* {{cite book |ref=harv |title=Linear Algebra |first1=S. |last1=Lipschutz |first2=M. |last2=Lipson |series=Schaum's Outline Series |year=2009 |isbn=978-0-07-154352-1}}
 
==External links==
 
*{{PlanetMath |urlname=DirectSumOfMatrices |title= Direct sum of matrices}}
 
* [http://ncalculators.com/matrix/4x4-matrix-addition-subtraction-calculator.htm 4x4 Matrix Addition and Subtraction]
* [http://drexel28.wordpress.com/2010/12/22/direct-sum-of-linear-transformations-and-direct-sum-of-matrices-pt-iii/ Abstract nonsense: Direct Sum of Linear Transformations and Direct Sum of Matrices]
* [http://www.mymathlib.com/matrices/arithmetic/direct_sum.html Mathematics Source Library: Arithmetic Matrix Operations]
* [http://www.aps.uoguelph.ca/~lrs/ABMethods/NOTES/CDmatrix.pdf Matrix Algebra and R]
 
[[Category:Linear algebra]]
[[Category:Binary operations]]

Revision as of 19:20, 3 March 2014

It is widely used to improve vision, strengthen bones and sinews and to improve overall sexual functions. In other words, these are drugs that cause impotence. Chances are, you are probably reading this because either you or a loved one suffers from male impotence. There are many reasons that business ethics matter. So its very important for you to have an awareness about the various signs of skin care that can help you detect skin cancer.

Endocrinologic Impotence: Endocrinologic impotence occurs due to an imbalance of sex hormones in the blood stream. Who think that even though we're writing about the internet, we should still make an effort to be entertaining and erudite. This medication may take up to four weeks to become fully effective, if used properly. Taken 1 hour before anticipated sexual activity, Viagra remains active in your body for about 4 hours. Lots of scientific findings confirm it helps increase immune system as well as improve level of energy.

In most cases the cause for male impotence is a physical problem. Overall, neither the man nor woman should become greatly concerned if they are infertile until one full year has passed with unprotected intercourse to get pregnant. For better results, half a teaspoon of ginger juice should be taken with a half boiled egg and honey, once daily at night for a month. If these key treatments do not assist you, your specialist may recommend you impotence drugs such as Viagra, Cialis and Levitra. mes, liquids and gels and want a fast, esthetically pleasing solution, this is the perfect product.

To treat heart disease you can take a small amount of black seed oil daily. Another popular herb is the Panax Ginseng, which is generally used for improving mental and physical prowess. All generic drugs are non prescription drugs and Forzest also belongs to the same category. PETA claims that meat eaters have higher rates of impotency due to artery blockage. L-arginine, a well known amino acid is another common cure suggested for reproductive problem.

The best way to avoid this is to always use photo release forms when shooting. Nerve fibers in any part of the body can become damaged in various ways. Laparoscopic RP and robotic RP are becoming increasingly more popular among patients for treatment of PCA. Of course, if your personal or family history suggests that no wine is best for you, then forego the vino and opt for red grape juice, which contains those same antioxidants. The Penis contains two chambers, made of spongy tissue, which run the length of the penis.

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