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matrix multiplication
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the multiplication of matrices
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Group-theoretic Algorithms for Matrix Multiplication. arXiv:math.GR/0511460.
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The operations of matrix addition and matrix multiplication are especially simple for symmetric diagonal matrices.
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matrix multiplication, matrix product
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matrix multiplication
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What is the fastest algorithm for matrix multiplication?
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The standard matrix multiplication takes approximately 2N3 (where N = 2n) arithmetic operations (additions and multiplications); the asymptotic complexity is Θ(N3).
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The convolution operation can be constructed as a matrix multiplication, where one of the inputs is converted into a Toeplitz matrix.
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Moreover, since composition of rotations corresponds to matrix multiplication, the rotation group is isomorphic to the special orthogonal group SO(3).
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By the formulas above, the n × n permutation matrices form a group under matrix multiplication with the identity matrix as the identity element.
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We still need 8 multiplications to calculate the Ci,j matrices, the same number of multiplications we need when using standard matrix multiplication.
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Practical implementations of Strassen's algorithm switch to standard methods of matrix multiplication for small enough submatrices, for which those algorithms are more efficient.
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They also showed that either of two different conjectures would imply that the optimal exponent of matrix multiplication is 2, as has long been suspected.
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In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication.
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If the matrix A has multiple eigenvalues, or is close to a matrix with multiple eigenvalues, then its Jordan normal form is very sensitive to perturbations.
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More generally, the general linear group of degree n over any field F (such as the complex numbers), or a ring R (such as the ring of integers), is the set of n×n invertible matrices with entries from F (or R), again with matrix multiplication as the group operation.
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Toeplitz matrices are also closely connected with Fourier series, because the multiplication operator by a trigonometric polynomial, compressed to a finite-dimensional space, can be represented by such a matrix.
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Consequently, if all singular values of a square matrix M are non-degenerate and non-zero, then its singular value decomposition is unique, up to multiplication of a column of U by a unit-phase factor and simultaneous multiplication of the corresponding column of V by the same unit-phase factor.
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For example, to solve a system of n equations for n unknowns by performing row operations on the matrix until it is in echelon form, and then solving for each unknown in reverse order, requires n(n + 1)/2 divisions, (2n3 + 3n2 − 5n)/6 multiplications, and (2n3 + 3n2 − 5n)/6 subtractions, for a total of approximately 2n3/3 operations.
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