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Since matrix multiplication forms the basis for many algorithms, and many operations on matrices even have the same complexity as matrix multiplication (up to a multiplicative constant), the computational complexity of matrix multiplication appears throughout numerical linear algebra and theoretical computer science.
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
In theoretical computer science, the computational complexity of matrix multiplication dictates how quickly the operation of matrix multiplication can be performed. Matrix multiplication algorithms are a central subroutine in theoretical and numerical algorithms for numerical linear algebra and optimization, so finding the fastest algorithm for matrix multiplication is of major practical ...
Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. [2] Over a field, a square matrix that is not invertible is called singular or degenerate. A square matrix with entries in a field is singular if and only if its determinant is zero.
The left column visualizes the calculations necessary to determine the result of a 2x2 matrix multiplication. Naïve matrix multiplication requires one multiplication for each "1" of the left column. Each of the other columns (M1-M7) represents a single one of the 7 multiplications in the Strassen algorithm. The sum of the columns M1-M7 gives ...
In 2005, Henry Cohn, Robert Kleinberg, Balázs Szegedy, and Chris Umans showed that either of two different conjectures would imply that the exponent of matrix multiplication is 2. [ 34 ] Transforms
The corresponding elementary matrix is a diagonal matrix, with diagonal entries 1 everywhere except in the i th position, where it is m. = [] So D i (m)A is the matrix produced from A by multiplying row i by m.
A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries. Matrices have a long history of both study and application, leading to diverse ways of classifying matrices. A first group is matrices satisfying concrete conditions of the entries, including constant matrices.