Seymour Lipschutz Marc Lipson Popular Books
Seymour Lipschutz Marc Lipson Biography & Facts
In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. For a vector, v→{\displaystyle {\vec {v}}\!}, adding two matrices would have the geometric effect of applying each matrix transformation separately onto v→{\displaystyle {\vec {v}}\!}, then adding the transformed vectors. Av→+Bv→=(A+B)v→{\displaystyle \mathbf {A} {\vec {v}}+\mathbf {B} {\vec {v}}=(\mathbf {A} +\mathbf {B} ){\vec {v}}\!}However, there are other operations that could also be considered addition for matrices, such as the direct sum and the Kronecker sum. Entrywise sum Two matrices must have an equal number of rows and columns to be added. In which case, the sum of two matrices A and B will be a matrix which has the same number of rows and columns as A and B. The sum of A and B, denoted A + B, is computed by adding corresponding elements of A and B: A+B=[a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮am1am2⋯amn]+[b11b12⋯b1nb21b22⋯b2n⋮⋮⋱⋮bm1bm2⋯bmn]=[a11+b11a12+b12⋯a1n+b1na21+b21a22+b22⋯a2n+b2n⋮⋮⋱⋮am1+bm1am2+bm2⋯amn+bmn]{\displaystyle {\begin{aligned}\mathbf {A} +\mathbf {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{aligned}}\,\!}Or more concisely (assuming that A + B = C): cij=aij+bij{\displaystyle c_{ij}=a_{ij}+b_{ij}}For example: [131012]+[007521]=[1+03+01+70+51+22+1]=[138533]{\displaystyle {\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}}}Similarly, it is also possible to subtract one matrix from another, as long as they have the same dimensions. The difference of A and B, denoted A − B, is computed by subtracting elements of B from corresponding elements of A, and has the same dimensions as A and B. For example: [131012]−[007521]=[1−03−01−70−51−22−1]=[13−6−5−11]{\displaystyle {\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}}}Direct sum Another operation, which is used less often, is the direct sum (denoted by ⊕). 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: A⊕B=[A00B]=[a11⋯a1n0⋯0⋮⋱⋮⋮⋱⋮am1⋯amn0⋯00⋯0b11⋯b1q⋮⋱⋮⋮⋱⋮0⋯0bp1⋯bpq]{\displaystyle \mathbf {A} \oplus \mathbf {B} ={\begin{bmatrix}\mathbf {A} &{\boldsymbol {0}}\\{\boldsymbol {0}}&\mathbf {B} \end{bmatrix}}={\begin{bmatrix}a_{11}&\cdots &a_{1n}&0&\cdots &0\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\a_{m1}&\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}}}For instance, [132231]⊕[1601]=[13200231000001600001]{\displaystyle {\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}}}The direct sum of matrices is a special type of block matrix. In particular, the direct sum of square matrices is a block diagonal matrix. The adjacency matrix of the union of disjoint graphs (or multigraphs) is the direct sum of their adjacency matrices. Any element in the direct sum of two vector spaces of matrices can be represented as a direct sum of two matrices. In general, the direct sum of n matrices is: ⨁i=1nAi=diag(A1,A2,A3,…,An)=[A10⋯00A2⋯0⋮⋮⋱⋮00⋯An]{\displaystyle \bigoplus _{i=1}^{n}\mathbf {A} _{i}=\operatorname {diag} (\mathbf {A} _{1},\mathbf {A} _{2},\mathbf {A} _{3},\ldots ,\mathbf {A} _{n})={\begin{bmatrix}\mathbf {A} _{1}&{\boldsymbol {0}}&\cdots &{\boldsymbol {0}}\\{\boldsymbol {0}}&\mathbf {A} _{2}&\cdots &{\boldsymbol {0}}\\\vdots &\vdots &\ddots &\vdots \\{\boldsymbol {0}}&{\boldsymbol {0}}&\cdots &\mathbf {A} _{n}\\\end{bmatrix}}\,\!}where the zeros are actually blocks of zeros (i.e., zero matrices). 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 Ik{\displaystyle \mathbf {I} _{k}} denotes the k-by-k identity matrix then the Kronecker sum is defined by: A⊕B=A⊗Im+In⊗B.{\displaystyle \mathbf {A} \oplus \mathbf {B} =\mathbf {A} \otimes \mathbf {I} _{m}+\mathbf {I} _{n}\otimes \mathbf {B} .}See also Matrix multiplication Vector additionNotes References Lipschutz, Seymour; Lipson, Marc (2017). Schaum's Outline of Linear Algebra (6 ed.). McGraw-Hill Education. ISBN 9781260011449. Riley, K.F.; Hobson, M.P.; Bence, S.J. (2006). Mathematical methods for physics and engineering (3 ed.). Cambridge University Press. doi:10.1017/CBO9780511810763. ISBN 978-0-521-86153-3.External links Direct sum of matrices at PlanetMath. Abstract nonsense: Direct Sum of Linear Transformations and Direct Sum of Matrices Mathematics Source Library: Arithmetic Matrix Operations Matrix Algebra and R. Discover the Seymour Lipschutz Marc Lipson popular books. Find the top 100 most popular Seymour Lipschutz Marc Lipson books.
Best Seller Seymour Lipschutz Marc Lipson Books of 2024
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My Paper Heart
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Saltwater Cove
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The Wonderful Wizard of Oz
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A Bookshop to Die For
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Pretty Little Lies
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Hot Off the Press
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Always Yours
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Holy Bible
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How to Choose a Guy in 10 Days
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The Four Loves
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Tempting the King
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Once Upon A One-Night Stand
Zoey LockeAt first sight, there was electrifying chemistry.  So why not go for it? After all, Lynx Grove, the city's most eligible bachelor, wants to claim her, at least for th...
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Coffee Girl
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Rogue Alpha
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Teach Me
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Nothing to Hide
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Good Guy
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His Girl Next Door
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All Fired Up
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Become A Better Version of Yourself
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Becoming Lady Dalton
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Inception of Gold
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The Scarlet Letter
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Enemies With Benefits
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The Seduction Series Boxset
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Death at Hazel House
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The Three Little Pigs
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Escape, A New Life
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Man In The Water
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The Icing on the Cake
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The Art of War
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The Count of Monte Cristo
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Hard Love
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Never Enough
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Law of Attraction
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Salvation
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Eternal
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The Stoic Mind
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Teased by Fire
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Just Me
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Bewitching a Highlander
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The Odyssey
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Silenced Girls
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The Cupcake Cottage
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The Adventures of Sherlock Holmes
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Get Lucky
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Dark Psychology and Manipulation
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Dracula
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Meditations
Emperor of Rome Marcus AureliusMeditations is a series of personal writings by Marcus Aurelius, Roman Emperor 161–180 CE, setting forth his ideas on Stoic philosophy.
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Caught Up with the Captain
Kait NolanCan a retired naval commander and the love he left behind overcome a 34yearold secret to find their way to a second chance? Captain Mitchell Greyson is a man who believes in duty. ...