Matrix Multiplication
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 - Michigan State University
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
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
Section 24 - Properties of Matrix-Matrix Multiplication
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
CS 140 : Matrix multiplication
Sequential Matrix Multiplication Simple mathematics, but getting good performance is complicated by memory hierarchy --- cache issues Naïve 3-loop matrix multiply
Matrix Multiplication - SageMath
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-vector Multiplication
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
2 Approximating Matrix Multiplication
2 2 Approximating matrix multiplication by random sampling We will start by considering a very simple randomized algorithm to approximate the product of two matrices Matrix multiplication is a fundamental linear algebraic problem, and this randomized algorithm for it is of interest in its own right
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