Matrix Multiplication Brian Krummel February 28, 2020 Today we want to de ne matrix multiplication The idea is to de ne matrix multiplication as a composition of linear transformations In particular, let A be an m n matrix and B be a n p matrix Let T A(X) = AX and T B(X) = BX be the corresponding matrix transformations We
Matrix Multiplication Simplify Write "undefined" for expressions that are undefined 1) 0 2 −2 −5 ⋅ 6 −6 3 0 2) 6 −3 ⋅ −5 4 3) −5 −5
Notes on Matrix Multiplication and the Transitive Closure Instructor: Sandy Irani An n m matrix over a set S is an array of elements from S with n rows and m columns Each element in a matrix is called an entry The entry in row i and column j is denoted by A i;j A matrix is called a square matrix if the number of rows is equal to the number
Matrix-Matrix Multiplication is Associative Let A, B, and C be matrices of conforming dimensions Then (AB)C = A(BC): Proof Let e j equal the jth unit basis vector Then (AB)Ce j = (AB)c
Sequential Matrix Multiplication Simple mathematics, but getting good performance is complicated by memory hierarchy --- cache issues Naïve 3-loop matrix multiply
J: matrix of Jordan blocks for eigenvalues P: nonsingular matrix A smith_form() triple with: D == U*A*V D: elementary divisors on diagonal U, V: with unit determinant A LU() triple with: P*A == L*U P: a permutation matrix L: lower triangular matrix, U: upper triangular matrix A QR() pair with: A == Q*R Q: a unitary matrix, R: upper triangular
matrix Sequential algorithm complexity: (n2) – multiplying n elements of each row of the matrix times n elements of the vector Parallel algorithm computational complexity: (n2/p) Communication complexity of all-gather: (log p + n) Why? All processes sending log p results to one process Assuming that p is a square number
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Les matrices - Propriétés de la multiplication
Cas particuliers de la multiplication Multiplication d’une matrice ligne et d’une matrice colonne Nous envisageons ici une multiplication de type (1 n)(n 1) (1 1) Dans ce cas, il n’y a qu’une seule ligne dans la première matrice et une seule colonne dans la seconde Et tout comme une matrice à une seule colonne est nommée vecteur colonne, une
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Les matrices - Multiplication - Clipedia
1 application de la loi de multiplication entre une matrice 2 2 et un vecteur colonne 2 1 (effectuer le produit) 2 idem 3 application de la loi de distributivité de la multiplication sur l’addition pour les sca-laires (distribuer d’abord, mettre en évidence in fine) et de la loi de commutativité de
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Chapitre 21 Matrices - maths-francefr
d) Pour toute matrice colonne U de format n, U +(−U) = 0 2) a) Pour toute matrice colonne U de format n et tous réels λ et µ, (λ+µ)U = λU +µU b) Pour toutes matrices colonnes U et V et tout réel λ, λ(U +V ) = λU +λV 2) Multiplication des matrices a) Multiplication d’un vecteur ligne de format n par un vecteur colonne de format n
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Matrix Multiplication - University of Plymouth
Section 2: Matrix Multiplication 1 6 2 Matrix Multiplication 1 The previous section gave the rule for the multiplication of a row vector A with a column vector B, the inner product AB This section will extend this idea to more general matrices Suppose that A = a 1 a 2 a n c 1 c 2 c n and B = (b 1 b 2 b n)T Then AB = a 1 a 2 a n c 1 c 2 c n b 1 b 2 ··Taille du fichier : 275KB
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Chapitre 6 : Matrices - e-monsite
1 Addition de matrices et multiplication d’un réel par une matrice Définition On définit les opérations suivantes sur l’ensemble Mnp(R) : Addition : ∀A = (aij)16i6n 16j6p ∈Mnp(R), ∀B = (bij)16i6n 16j6p ∈Mnp(R), A +B = (aij +bij)16i6n 16j6p ∈Mnp(R) Multiplication par un réel : ∀λ ∈R, ∀A = (aij)16i6n 16j6p ∈Mnp(R), λA = (λaij)16i6n 16j6p
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Multiplication rapide de polynômes et de matrices
soustr : matrice -> matrice -> matrice qui calculent respectivement la somme de deux matrices, la multiplication d'une matrice par un scalaire, et la di érence de deux matrices On supposera que les matrices sont bien formées (toutes les lignes ont la même longueur), et que les dimensions sont compatibles 2 2 Multiplication naïve Si A = (a
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Synthèse 3 : Les matrices
2 2 Multiplication d’une matrice par un scalaire Définition Soient A = aij une matrice de dimension (np,) et λ∈\ On définit la matrice λA comme matrice dont tous les coefficients sont multipliés par λ : λA = λaij λA est aussi de dimension ()np, Propriétés
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Opérations sur les matrices - unicefr
q colonnes Voici la carte de visite de la multiplication externe des matrices, qu’on appelle encore Multex Multex p,q: R×M p,q → M p,q (λ,A) 7→ λA (λ,A) 7→ (i,j) 7→λA ij Taille du fichier : 159KB
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Exo7 - Cours de mathématiques
L’addition et la multiplication par un scalaire se comportent sans surprises : Proposition 1 Soient A, B et C trois matrices appartenant à Mn,p(K) Soient 2K et 2K deux scalaires 1 A+B = B +A : la somme est commutative, 2 A+(B +C) = (A+B)+C : la somme est associative, 3 A+0 = A : la matrice nulle est l’élément neutre de l’addition, 4 ( + )A= A+ A, 5 Taille du fichier : 220KB
We then present a second matrix multiplication algorithm which is similar in spirit to our main algorithm. In addition we present a model (the pass-efficient
approach” to implementing matrix multiplication (GEMM). While. GEMM was previously implemented as three loops around an inner kernel BLIS exposes two
in distributed matrix multiplication. Furthermore by leveraging the algebraic structure of polynomial codes
Extra memory allows parallel matrix multiplication to be done with asymptotically less communication than Cannon's algorithm and be faster in practice.
matrix multiplication that is part of the widely used GotoBLAS library. Design decisions are justified by successively refining a model of architectures
Additional Key Words and Phrases: linear algebra matrix multiplication
Approximating the product of two matrices with random sampling or random projection methods is a fundamental operation that is of interest in and of itself
Unlike conventional methods using inner or outer product as the meta operation for matrix multiplication our approach is based on row-wise product
arithmetic intensity matrix multiplication
Jun 11 2012 Abstract. We describe an extension of the Scalable Universal Matrix Multiplication Algorithms (SUMMA) from 2D to 3D process grids; ...
There is one very important property of matrix multiplication that it is best to see early on Considerthe calculation below in which two square matrices are multiplied in a di?erent order 1 2 3 2 ?11 ?1 5 5 = 35 ?5 3 ?1 1 2 1 7 = 32 ?17 ?1 We see from this that matrix multiplication is not commutative
Algebra of Matrix Multiplication Identity Matrix Number of Solutions Properties of Matrix Multiplication Let A;B;C be matrices and c is a constant Assume all the matrix products below are de ned Then A(BC) = (AB)C Associativity Matrix Product A(B + C) = AB + AC Distributive Property (A+ B)C = AC + BC Distributive Property c(AB) = (cA)B = A(cB)
Matrix algebra: linear operations Addition: two matrices of the same dimensionscan be added by adding their corresponding entries Scalar multiplication: to multiply a matrixAby scalarr one multiplies each entry of Abyr Zero matrixO: all entries are zeros Negative: ?Ais de?ned as (?1)A Subtraction: A?Bis de?ned asA+ (?B)
Chapter 2 Matrices and Linear Algebra 2 1 Basics De?nition 2 1 1 A matrix is an m×n array of scalars from a given ?eld F The individual values in the matrix are called entries
The method of multiplication of matrices is not asintuitive and may seem strange although this methodis extremely useful in many mathematical applications Matrix multiplication was introduced by an Englishmathematician named Arthur Cayley (1821-1895) We will see shortly how matrix multiplication can beused to solve systems of linear
Multiplication Just like adding and subtracting we first need to take a look at the size of the two matrices we want to multiply Matrix A Matrix B The number of columns in the first matrix MUST be the same as the number of rows in the second matrix otherwise the answer is “undefined”
What is a solution using matrix multiplication?
Solution using matrix multiplication ?We represent the number of each model sold using a row matrix (4X1) and we use a 1X4 column matrix to represent the sales price of each model. When a 4X1 matrix is multiplied by a 1X4 matrix, the result is a 1X1 matrix of a single number.
How many matrix multiplications are there?
0 0 2Note there are two matrix multiplications them, one for each Type 3 ele-mentary operation. by row operations. Called theRREF, it has the following properties. Each nonzero row has a 1 as the?rst nonzero entry (:=leading one). (b) All column entries above and below a leading one are zero.
Who invented matrix multiplication?
?Matrix multiplication was introduced by an English mathematician named Arthur Cayley (1821-1895) . ?We will see shortly how matrix multiplication can be used to solve systems of linear equations. Row by column multiplication
How do you multiply two matrices?
Just like adding and subtracting, we first need to take a look at the size of the two matrices we want to multiply. The number of columns in the first matrix MUST be the same as the number of rows in the second matrix, otherwise, the answer is “undefined”. same number of columns as the second matrix. tricky.
FAST MONTE CARLO ALGORITHMS FOR MATRICES I