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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]].
| | Im Taylor and was born on 10 March 1986. My hobbies are Auto audiophilia and Stamp collecting.<br><br>My web-site - [http://download.cnet.com/PDF-to-Word-Converter-Free/3000-18497_4-76126166.html Free PDF to Word Converter] |
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| ==Entrywise sum==
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| 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>
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| :<math>\begin{align}
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| \bold{A}+\bold{B} & = \begin{bmatrix}
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| a_{11} & a_{12} & \cdots & a_{1n} \\
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| a_{21} & a_{22} & \cdots & a_{2n} \\
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| \vdots & \vdots & \ddots & \vdots \\
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| a_{m1} & a_{m2} & \cdots & a_{mn} \\
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| \end{bmatrix} +
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| \begin{bmatrix}
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| b_{11} & b_{12} & \cdots & b_{1n} \\
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| b_{21} & b_{22} & \cdots & b_{2n} \\
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| \vdots & \vdots & \ddots & \vdots \\
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| b_{m1} & b_{m2} & \cdots & b_{mn} \\
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| \end{bmatrix} \\
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| & = \begin{bmatrix}
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| a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n} \\
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| a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n} \\
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| \vdots & \vdots & \ddots & \vdots \\
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| a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots & a_{mn} + b_{mn} \\
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| \end{bmatrix} \\
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| \end{align}\,\!</math>
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| For example:
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| :<math>
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| \begin{bmatrix}
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| 1 & 3 \\
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| 1 & 0 \\
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| 1 & 2
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| \end{bmatrix}
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| +
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| \begin{bmatrix}
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| 0 & 0 \\
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| 7 & 5 \\
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| 2 & 1
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1+0 & 3+0 \\
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| 1+7 & 0+5 \\
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| 1+2 & 2+1
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1 & 3 \\
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| 8 & 5 \\
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| 3 & 3
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| \end{bmatrix}
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| </math>
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| We can also subtract one matrix from another, as long as they have the same dimensions. '''A''' − '''B''' is computed by subtracting corresponding elements of '''A''' and '''B''', and has the same dimensions as '''A''' and '''B'''. For example:
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| :<math>
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| \begin{bmatrix}
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| 1 & 3 \\
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| 1 & 0 \\
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| 1 & 2
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| \end{bmatrix}
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| -
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| \begin{bmatrix}
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| 0 & 0 \\
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| 7 & 5 \\
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| 2 & 1
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1-0 & 3-0 \\
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| 1-7 & 0-5 \\
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| 1-2 & 2-1
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1 & 3 \\
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| -6 & -5 \\
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| -1 & 1
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| \end{bmatrix}
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| </math>
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| ==<span id="directsum" />Direct sum==
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| 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'' × ''n'' and '''B''' of size ''p'' × ''q'' is a matrix of size (''m'' + ''p'') × (''n'' + ''q'') defined as <ref>{{MathWorld |id=MatrixDirectSum |title=Matrix Direct Sum}}</ref>{{sfn |Lipschutz |Lipson}}
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| :<math>
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| \bold{A} \oplus \bold{B} =
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| \begin{bmatrix} \bold{A} & \boldsymbol{0} \\ \boldsymbol{0} & \bold{B} \end{bmatrix} =
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| \begin{bmatrix}
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| a_{11} & \cdots & a_{1n} & 0 & \cdots & 0 \\
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| \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
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| a_{m 1} & \cdots & a_{mn} & 0 & \cdots & 0 \\
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| 0 & \cdots & 0 & b_{11} & \cdots & b_{1q} \\
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| \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
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| 0 & \cdots & 0 & b_{p1} & \cdots & b_{pq}
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| \end{bmatrix}
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| </math>
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| For instance,
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| :<math>
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| \begin{bmatrix}
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| 1 & 3 & 2 \\
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| 2 & 3 & 1
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| \end{bmatrix}
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| \oplus
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| \begin{bmatrix}
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| 1 & 6 \\
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| 0 & 1
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1 & 3 & 2 & 0 & 0 \\
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| 2 & 3 & 1 & 0 & 0 \\
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| 0 & 0 & 0 & 1 & 6 \\
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| 0 & 0 & 0 & 0 & 1
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| \end{bmatrix}
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| </math>
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| 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]].
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| 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.
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| In general, the direct sum of ''n'' matrices is:{{sfn |Lipschutz |Lipson}}
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| :<math>
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| \bigoplus_{i=1}^{n} \bold{A}_{i} = {\rm diag}( \bold{A}_1, \bold{A}_2, \bold{A}_3 \cdots \bold{A}_n)=
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| \begin{bmatrix}
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| \bold{A}_1 & \boldsymbol{0} & \cdots & \boldsymbol{0} \\
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| \boldsymbol{0} & \bold{A}_2 & \cdots & \boldsymbol{0} \\
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| \vdots & \vdots & \ddots & \vdots \\
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| \boldsymbol{0} & \boldsymbol{0} & \cdots & \bold{A}_n \\
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| \end{bmatrix}\,\!</math>
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| where the zeros are actually blocks of zeros, i.e. zero matricies.
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| NB: Sometimes in this context, boldtype for matrices is dropped, matricies are written in italic.
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| ==Kronecker sum==
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| {{main|Kronecker sum}}
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| 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:
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| :<math> \mathbf{A} \oplus \mathbf{B} = \mathbf{A} \otimes \mathbf{I}_m + \mathbf{I}_n \otimes \mathbf{B}. </math>
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| ==See also==
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| * [[Matrix multiplication]]
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| * [[Vector addition]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{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}}
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| ==External links==
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| *{{PlanetMath |urlname=DirectSumOfMatrices |title= Direct sum of matrices}}
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| * [http://ncalculators.com/matrix/4x4-matrix-addition-subtraction-calculator.htm 4x4 Matrix Addition and Subtraction]
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| * [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]
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| * [http://www.mymathlib.com/matrices/arithmetic/direct_sum.html Mathematics Source Library: Arithmetic Matrix Operations]
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| * [http://www.aps.uoguelph.ca/~lrs/ABMethods/NOTES/CDmatrix.pdf Matrix Algebra and R]
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| [[Category:Linear algebra]]
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| [[Category:Binary operations]]
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Im Taylor and was born on 10 March 1986. My hobbies are Auto audiophilia and Stamp collecting.
My web-site - Free PDF to Word Converter