Matrix Multiplication

Matrix multiplicationFrom Wikipedia, the free encyclopediaIn mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces anothermatrix. Numbers such as the real or complex numbers can be multiplied according to elementary arithmetic. Onthe other hand, matrices are arrays of numbers, so there is no unique way to define "the" multiplication ofmatrices. As such, in general the term "matrix multiplication" refers to a number of different ways to multiplymatrices. The key features of any matrix multiplication include: the number of rows and columns the originalmatrices have (called the "size", "order" or "dimension"), and specifying how the entries of the matrices generatethe new matrix.Like vectors, matrices of any size can be multiplied by scalars, which amounts to multiplying every entry of thematrix by the same number. Similar to the entrywise definition of adding or subtracting matrices, multiplicationof two matrices of the same size can be defined by multiplying the corresponding entries, and this is known as theHadamard product.One can form many other definitions. However, the most useful definition can be motivated by linear equationsand linear transformations on vectors, which have numerous applications in applied mathematics, physics, andengineering. This definition is often called the matrix product.[1][2] Although this definition is not commutative, itstill retains the associative property and is distributive over entrywise addition of matrices, and leads to thedefinitions of: an identity matrix (analogous to multiplying real numbers by the number 1), an inverse matrix(analogous to the multiplicative inverse of a number), powers and nth roots of a square matrix, consequently thematrix exponential can be defined by a power series of a square matrix, and so on. A consequence of this matrixproduct is determinant multiplicativity. This matrix product is an important operation in matrix groups, and thetheory of group representations and irreps.This article will use the following notational conventions: matrices are represented by capital letters in bold,vectors in lowercase bold, and entries of vectors and matrices are italic (since they are scalars). Index notation isoften the clearest way to express definitions, and will be used as standard in the literature.

Contents1 Scalar multiplication

1.1 General definition1.2 Examples

2 Matrix product (two matrices)2.1 General definition2.2 Illustration2.3 Examples

3 Properties of matrix multiplication3.1 General

3.1.1 All matrices3.1.2 Square matrices only

3.2 Linear transformations4 Matrix product (any number)

4.1 Chain multiplication4.1.1 General definition

4.2 Powers of matrices4.3 Powers of diagonal matrices

5 The inner and outer products5.1 Examples

6 Algorithms for efficient matrix multiplication6.1 Communication-avoiding and distributed algorithms

7 Other forms of multiplication7.1 Hadamard product7.2 Frobenius product

Matrix multiplication - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Matrix_multiplication

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7.3 Kronecker product8 See also9 Notes10 References11 External links

Scalar multiplicationMain article: Scalar multiplication

The simplest form of multiplication associated with matrices is scalar multiplication.General definitionLeft scalar multiplicationThe left multiplication of a matrix A with a scalar gives another matrix A of the same size as A. The entries ofA are given by

explicitly:

Right scalar multiplicationSimilarly, the right multiplication of a matrix A with a scalar is defined to be

When the underlying ring is commutative, for example, the real or complex number field, these twomultiplications are the same, and are simply called scalar multiplication. However, for matrices over a moregeneral ring that are not commutative, such as the quaternions, they may not be equal.ExamplesFor a real scalar and matrix:

For quaternion scalars and matrices:

Matrix product (two matrices)


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Arithmetic process of multiplying numbers(solid lines) in row i in matrix A andcolumn j in matrix B, then adding the terms(dashed lines) to obtain entry ij in the finalmatrix.

Assume two matrices are to be multiplied (the generalization to any number is discussed below). If A is an nmmatrix and B is an mp matrix, the result AB of their multiplication is an np matrix defined only if the number ofcolumns m in A is equal to the number of rows m in B.General definitionWhen multiplying matrices, the elements of the rows in the firstmatrix are multiplied with corresponding columns in the secondmatrix (depicted in the image right). One may compute each entry inthe third matrix one at a time.For two matrices

where necessarily the number of columns in A equals the number of rows in B, in this case m, the matrix productAB is denoted without symbol (no multiplication signs or dots) to be the np matrix:[3][4]

where AB has entries defined by:

Treating the rows and columns in each matrix as row and column vectors respectively, this entry is also theirvector dot product:

(See below for further details). Usually the entries are numbers or expressions, but can even be matricesthemselves (see block matrix). The matrix product can still be calculated exactly the same way.


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IllustrationThe figure to the right illustrates diagrammatically the product of twomatrices A and B, showing how each intersection in the productmatrix corresponds to a row of A and a column of B.

The values at the intersections marked with circles are:

ExamplesSquare matrix and column vectorIf

their matrix product is:

yet BA is not defined.The product of a square matrix multiplied by a column matrix arises naturally in linear algebra; for solving linearequations and representing linear transformations. By choosing a, b, c, d in A appropriately, A can represent avariety of transformations such as rotations, scaling and reflections, shears, of a geometric shape in space.Square matricesIf

their matrix products are:

and

In this case, both products AB and BA are defined, and the entries show that AB and BA are not equal in general.Multiplying square matrices which represent linear transformations corresponds to the composite transformation(see below for details). Also, similarity transformations involving similar matrices are matrix products of squarematrices, which requires the notion of an inverse matrix (see also below).


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Properties of matrix multiplicationGeneralAnalogous to numbers (elements of a field), matrices satisfy the following general properties, although there is onesubtlety, due to the nature of matrix multiplication.All matrices

Not commutative:In general:because AB and BA may not be simultaneously defined, and even if they are they may still not be equal. Thisis contrary to ordinary multiplication of numbers. To specify the ordering of matrix multiplication in words;"pre-multiply (or left multiply) A by B" means BA, while "post-multiply (or right multiply) A by C" means AC.As long as the entries of the matrix come from a ring that has an identity, and n > 1 there is a pair of nnnoncommuting matrices over the ring. A notable exception is that the identity matrix (or any scalar multipleof it) commutes with every square matrix.

1.

Associative:2.Distributive over matrix addition:3.Scalar multiplication is compatible with matrix multiplication:

and where is a scalar. If the entries of the matrix are real or complex numbers (or from any other commutativering), then all four quantities are equal. More generally, all four are equal if belongs to the center of the ringof entries of the matrix, because in this case X = X for all matrices X.

4.

Transpose:

where T denotes the transpose, the interchange of row i with column i in a matrix. This identity holds for anymatrices over a commutative ring, but not for all rings in general. Note that A and B are reversed.

5.

Hermitian conjugate: If A and B have complex entries, then

where denotes the Hermitian conjugate of a matrix (complex conjugate and transposed).

6.

Traces: The trace of a product AB is independent of the order of A and B:7.

Square matrices onlyMain article: square matrix

Identity element: If A is a square matrix, thenwhere I is the identity matrix of the same order.

1.

Inverse matrix: If A is a square matrix, there may be an inverse matrix A1 of A such thatIf this property holds then A is an invertible matrix, if not A is a singular matrix. Moreover,

2.

Determinants: The determinant of a product AB is the product of the determinants of square matrices A andB (not defined when the underlying ring is not commutative):Since det(A) and det(B) are just numbers and so commute, det(AB) = det(A)det(B) = det(B)det(A) = det(BA),even when AB BA.

3.

Linear transformations


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Main article: Linear transformationsMatrices offer a concise way of representing linear transformations between vector spaces, and matrixmultiplication corresponds to the composition of linear transformations. The matrix product of two matrices canbe defined when their entries belong to the same ring, and hence can be added and multiplied.Let U, V, and W be vector spaces over the same field with given bases, S: V W and T: U V be lineartransformations and ST: U W be their composition.Suppose that A, B, and C are the matrices representing the transformations S, T, and ST with respect to the givenbases.Then AB = C, that is, the matrix of the composition (or the product) of linear transformations is the product of theirmatrices with respect to the given bases.

Matrix product (any number)Chain multiplication

Main article: Matrix chain multiplicationMatrix multiplication can be extended to the case of more than two matrices, provided that for each sequentialpair, their dimensions match.General definitionThe product of N matrices A1, A2, ..., AN with sizes n0n1, n1n2, ..., nN 1nN, is the n0nN matrix:

The same properties will hold, as long as the ordering of matrices is not changed. The number of possible ways ofgrouping n matrices for multiplication is equal to the (n 1)th Catalan number.For example, if A, B, C, and D are respectively mp, pq, qr, and rn matrices, then there are 5 ways of groupingthem without changing their order, and

is an mn matrix.Powers of matricesSquare matrices can be multiplied by themselves repeatedly in the same way as ordinary numbers, because theyalways have the same number of rows and columns. This repeated multiplication can be described as a power ofthe matrix, a special case of the ordinary matrix product. On the contrary, rectangular matrices do not have thesame number of rows and columns so they can never be raised to a power. An nn matrix A raised to a positiveinteger k is defined as

and the following identities hold, where is a scalar:Zero power

where I is the identity matrix. This is parallel to the zeroth power of any number which equals unity.


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Scalar multiplication

Determinant

The naive computation of matrix powers is to multiply k times the matrix A to the result, starting with the identitymatrix just like the scalar case. This can be improved using exponentiation by squaring, a method commonly usedfor scalars. For diagonalizable matrices, an even better method is to use the eigenvalue decomposition of A.Another method based on the CayleyHamilton theorem finds an identity using the matrices' characteristicpolynomial, producing a more effective equation for Ak in which a scalar is raised to the required power, ratherthan an entire matrix.Powers of diagonal matricesA special case is the power of a diagonal matrix A.Since the product of diagonal matrices amounts to simply multiplying corresponding diagonal elements together,the power k of a diagonal matrix A will have entries raised to the power. Explicitly;

meaning it is easy to raise a diagonal matrix to a power. When raising an arbitrary matrix (not necessarily adiagonal matrix) to a power, it is often helpful to exploit this property by diagonalizing the matrix first.

The inner and outer productsGiven two column vectors a and b, the Euclidean inner product and outer product are the simplest special cases ofthe matrix product, by transposing the column vectors into row vectors.[5]

The inner productis a column vector multiplied on the left by a row vector:

More explicitly,

The outer productis a row vector multiplied on the left by a column vector:

where


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Matrix product (in terms of inner product)Suppose that the first nm matrix A is decomposed into its row vectors ai, and the second mp matrix B into itscolumn vectors bi:[1]

where

The entries in the introduction were given by:

It is also possible to express a matrix product in terms of concatenations of products of matrices and row orcolumn vectors:

These decompositions are particularly useful for matrices that are envisioned as concatenations of particulartypes of row vectors or column vectors, e.g. orthogonal matrices (whose rows and columns are unit vectorsorthogonal to each other) and Markov matrices (whose rows or columns sum to 1).Matrix product (in terms of outer product)An alternative method results when the decomposition is done the other way around, i.e. the first matrix A isdecomposed into column vectors ai and the second matrix B into row vectors bi:

where this time


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What is the fastest algorithm formatrix multiplication?

List of unsolved problems incomputer science

The bound on over time.

This method emphasizes the effect of individual column/row pairs on the result, which is a useful point of viewwith e.g. covariance matrices, where each such pair corresponds to the effect of a single sample point.ExamplesSuppose

using the inner product approach:

while the outer product approach gives:

Algorithms for efficient matrix multiplicationThe running time of square matrix multiplication, if carried out navely, is

. The running time for multiplying rectangular matrices (onemp-matrix with one pn-matrix) is , however, more efficientalgorithms exist, such as Strassen's algorithm, devised by Volker Strassen in1969 and often referred to as "fast matrix multiplication". It is based on away of multiplying two 22-matrices which requires only7 multiplications (instead of the usual 8), at the expenseof several additional addition and subtraction operations.Applying this recursively gives an algorithm with amultiplicative cost of . Strassen'salgorithm is more complex, and the numerical stability isreduced compared to the nave algorithm.[6]Nevertheless, it appears in several libraries, such asBLAS, where it is significantly more efficient for matriceswith dimensions n > 100,[7] and is very useful for largematrices over exact domains such as finite fields, wherenumerical stability is not an issue.The current algorithm with the lowest knownexponent k is a generalization of the CoppersmithWinograd algorithm that has an asymptotic complexity ofO(n2.3727) thanks to Vassilevska Williams.[8] Thisalgorithm, and the Coppersmith-Winograd algorithm onwhich it is based, are similar to Strassen's algorithm: away is devised for multiplying two kk-matrices withfewer than k3 multiplications, and this technique isapplied recursively. However, the constant coefficienthidden by the Big O notation is so large that these algorithms are only worthwhile for matrices that are too large to


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Block matrix multiplication. Inthe 2D algorithm, eachprocessor is responsible for onesubmatrix of C. In the 3Dalgorithm, every pair ofsubmatrices from A and B thatis multiplied is assigned to oneprocessor.

handle on present-day computers.[9]

Since any algorithm for multiplying two nn-matrices has to process all 2n2-entries, there is an asymptotic lowerbound of operations. Raz (2002) proves a lower bound of for bounded coefficient arithmeticcircuits over the real or complex numbers.Cohn et al. (2003, 2005) put methods such as the Strassen and CoppersmithWinograd algorithms in an entirelydifferent group-theoretic context, by utilising triples of subsets of finite groups which satisfy a disjointnessproperty called the triple product property (TPP). They show that if families of wreath products of Abelian groupswith symmetric groups realise families of subset triples with a simultaneous version of the TPP, then there arematrix multiplication algorithms with essentially quadratic complexity. Most researchers believe that this isindeed the case.[10] However, Alon, Shpilka and Umans have recently shown that some of these conjecturesimplying fast matrix multiplication are incompatible with another plausible conjecture, the sunflowerconjecture.[11]

Because of the nature of matrix operations and the layout of matrices in memory, it is typically possible to gainsubstantial performance gains through use of parallelization and vectorization. It should therefore be noted thatsome lower time-complexity algorithms on paper may have indirect time complexity costs on real machines.Freivalds' algorithm is a simple Monte Carlo algorithm that given matrices verifies in time if

.Communication-avoiding and distributed algorithmsOn modern architectures with hierarchical memory, the cost of loading andstoring input matrix elements tends to dominate the cost of arithmetic. On asingle machine this is the amount of data transferred between RAM and cache,while on a distributed memory multi-node machine it is the amount transferredbetween nodes; in either case it is called the communication bandwidth. Thenave algorithm using three nested loops uses communication bandwidth.Cannon's algorithm, also known as the 2D algorithm, partitions each input matrixinto a block matrix whose elements are submatrices of size by ,where M is the size of fast memory.[12] The nave algorithm is then used over theblock matrices, computing products of submatrices entirely in fast memory. Thisreduces communication bandwidth to , which is asymptoticallyoptimal (for algorithms performing computation).[13][14]

In a distributed setting with p processors arranged in a by 2D mesh, onesubmatrix of the result can be assigned to each processor, and the product can becomputed with each processor transmitting words, which isasymptotically optimal assuming that each node stores the minimum elements.[14] This can be improved by the 3D algorithm, which arranges theprocessors in a 3D cube mesh, assigning every product of two input submatricesto a single processor. The result submatrices are then generated by performing areduction over each row.[15] This algorithm transmits words perprocessor, which is asymptotically optimal.[14] However, this requires replicatingeach input matrix element times, and so requires a factor of morememory than is needed to store the inputs. This algorithm can be combined withStrassen to further reduce runtime.[15] "2.5D" algorithms provide a continuoustradeoff between memory usage and communication bandwidth.[16]

Other forms of multiplicationThere are other ways to multiply two matrices, in fact simpler than the definition above.


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Hadamard productMain article: Hadamard product (matrices)

For two matrices of the same dimensions, there is the Hadamard product, also known as the element-wiseproduct, pointwise product, entrywise product and the Schur product.[17] For two matrices A and B of thesame dimensions, the Hadamard product A B is a matrix of the same dimensions, which has elements

explicitly:

Due to the characteristic entrywise procedure, this operation is identical to many multiplying ordinary numbers(mn of them) all at once; hence the Hadamard product is commutative, associative and distributive over addition,and is a principal submatrix of the Kronecker product. It appears in lossy compression algorithms such as JPEG.Frobenius productThe Frobenius inner product, sometimes denoted A : B, is the component-wise inner product of two matrices asthough they are vectors. It is also the sum of the entries of the Hadamard product. Explicitly,

where "tr" denotes the trace of a matrix and vec denotes vectorization. This inner product induces the Frobeniusnorm.Kronecker product

Main article: Kronecker productFor two matrices A and B of any different dimensions mn and pq respectively (no constraints on the dimensionsof each matrix), the Kronecker product denoted A B is a matrix with dimensions mpnq, which haselements[citation needed]

,where represents the floor function.Explicitly:

This is the application of the more general tensor product applied to matrices.

See also


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Basic Linear AlgebraSubprogramsComposition of relations

CoppersmithWinogradalgorithmCracovianLogical matrix

Matrix additionMatrix exponentiationMatrix inversionStrassen algorithm

Notes^ a b Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft)3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3

1.^ McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-32.^ Linear Algebra (4th Edition), S. Lipcshutz, M. Lipson, Schaum's Outlines, McGraw Hill (USA), 2009, ISBN978-0-07-154352-1

3.^ Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010,ISBN 978-0-521-86153-3

4.^ Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press,2010, ISBN 978-0-521-86153-3

5.^ Miller, Webb (1975), "Computational complexity and numerical stability" (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.148.9947&rep=rep1&type=pdf), SIAM News 4: 97107

6.^ Press 2007, p. 108.7.^ Virginia Vassilevska Williams. "Multiplying matrices faster than Coppersmith-Winograd" (http://www.cs.stanford.edu/~virgi/matrixmult-f.pdf). The original algorithm was presented by Don Coppersmith and Shmuel Winograd in 1990, hasan asymptotic complexity of O(n2.376).

8.

^ Robinson, Sara (2005), "Toward an Optimal Algorithm for Matrix Multiplication" (http://www.siam.org/pdf/news/174.pdf), SIAM News 38 (9)

9.^ Robinson, 2005.10.^ Alon, Shpilka, Umans, On Sunflowers and Matrix Multiplication (http://eccc.hpi-web.de/report/2011/067/)11.^ Lynn Elliot Cannon, A cellular computer to implement the Kalman Filter Algorithm (http://portal.acm.org/citation.cfm?coll=GUIDE&dl=GUIDE&id=905686), Technical report, Ph.D. Thesis, Montana State University, 14 July 1969.

12.^ Hong, J.W.; H.T. Kung (1981). "I/O complexity: The red-blue pebble game". STOC 81: Proceedings of the thirteenth annualACM symposium on Theory of computing: 326333.

13.

^ a b c Irony, Dror; Sivan Toledo, Alexander Tiskin (September 2004). "Communication lower bounds for distributed-memory matrix multiplication". J. Parallel Distrib. Comput. 64 (9): 10171026. doi:10.1016/j.jpdc.2004.03.021(http://dx.doi.org/10.1016%2Fj.jpdc.2004.03.021).

14.

^ a b Agarwal, R.C.; S. M. Balle, F. G. Gustavson, M. Joshi, P. Palkar (September 1995). "A three-dimensional approach toparallel matrix multiplication". IBM J. Res. Dev. 39 (5): 575582. doi:10.1147/rd.395.0575 (http://dx.doi.org/10.1147%2Frd.395.0575).

15.

^ Solomonik, Edgar; James Demmel (2011). "Communication-optimal parallel 2.5D matrix multiplication and LUfactorization algorithms". Proceedings of the 17th international conference on Parallel processing. Part II: 90109.

16.^ (Horn & Johnson 1985,Ch. 5)17.

ReferencesHenry Cohn, Robert Kleinberg, Balazs Szegedy, and Chris Umans. Group-theoretic Algorithms for MatrixMultiplication. arXiv:math.GR/0511460. Proceedings of the 46th Annual Symposium on Foundations ofComputer Science, 2325 October 2005, Pittsburgh, PA, IEEE Computer Society, pp. 379388.Henry Cohn, Chris Umans. A Group-theoretic Approach to Fast Matrix Multiplication.arXiv:math.GR/0307321. Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science,1114 October 2003, Cambridge, MA, IEEE Computer Society, pp. 438449.Coppersmith, D., Winograd S., Matrix multiplication via arithmetic progressions, J. Symbolic Comput. 9,p. 251-280, 1990.Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press,ISBN 978-0-521-38632-6Horn, Roger A.; Johnson, Charles R. (1991), Topics in Matrix Analysis, Cambridge University Press,ISBN 978-0-521-46713-1Knuth, D.E., The Art of Computer Programming Volume 2: Seminumerical Algorithms. Addison-WesleyProfessional; 3 edition (November 14, 1997). ISBN 978-0-201-89684-8. pp. 501.Press, William H.; Flannery, Brian P.; Teukolsky, Saul A.; Vetterling, William T. (2007), Numerical Recipes: The


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Art of Scientific Computing (3rd ed.), Cambridge University Press, ISBN 978-0-521-88068-8.Ran Raz. On the complexity of matrix product. In Proceedings of the thirty-fourth annual ACM symposium onTheory of computing. ACM Press, 2002. doi:10.1145/509907.509932 (http://dx.doi.org/10.1145%2F509907.509932).Robinson, Sara, Toward an Optimal Algorithm for Matrix Multiplication, SIAM News 38(9), November 2005.PDF (http://www.siam.org/pdf/news/174.pdf)Strassen, Volker, Gaussian Elimination is not Optimal, Numer. Math. 13, p. 354-356, 1969.Styan, George P. H. (1973), "Hadamard Products and Multivariate Statistical Analysis", Linear Algebra and itsApplications 6: 217240, doi:10.1016/0024-3795(73)90023-2 (http://dx.doi.org/10.1016%2F0024-3795%2873%2990023-2)Vassilevska Williams, Virginia, Multiplying matrices faster than Coppersmith-Winograd, Manuscript, May2012. PDF (http://www.cs.stanford.edu/~virgi/matrixmult-f.pdf)

External linksHow to Multiply Matrices (http://www.mathsisfun.com/algebra/matrix-multiplying.html)The Simultaneous Triple Product Property and Group-theoretic Results for the Exponent of MatrixMultiplication (http://arxiv.org/abs/cs/0703145)WIMS Online Matrix Multiplier (http://wims.unice.fr/~wims/en_tool~linear~matmult.html)Matrix Multiplication Problems (http://ceee.rice.edu/Books/LA/mult/mult4.html#TOP)Block Matrix Multiplication Problems (http://www.gordon-taft.net/MatrixMultiplication.html)Wijesuriya, Viraj B., Daniweb: Sample Code for Matrix Multiplication using MPI Parallel ProgrammingApproach (http://www.daniweb.com/forums/post1428830.html#post1428830), retrieved 2010-12-29Linear algebra: matrix operations (http://www.umat.feec.vutbr.cz/~novakm/algebra_matic/en) Multiply oradd matrices of a type and with coefficients you choose and see how the result was computed.Visual Matrix Multiplication (http://www.wefoundland.com/project/Visual_Matrix_Multiplication) Aninteractive app for learning matrix multiplication.Matrix Multiplication in Java Dr. P. Viry (http://www.ateji.com/px/whitepapers/Ateji%20PX%20MatMult%20Whitepaper%20v1.2.pdf?phpMyAdmin=95wsvAC1wsqrAq3j,M3duZU3UJ7)

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Matrix Multiplication

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