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In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the ...
Elementary algebra. Propositional calculus. In mathematics, the associative property[1] is a property of some binary operations that means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.
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: Input: matrices A and B.
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 ...
An associative algebra is a ring that is also a vector space over a field n such that the scalar multiplication is compatible with the ring multiplication. For instance, the set of n -by- n matrices over the real field R {\displaystyle \mathbb {R} } has dimension n 2 as a real vector space.
Matrix multiplication also does not necessarily obey the cancellation law. If AB = AC and A ≠ 0, then one must show that matrix A is invertible (i.e. has det(A) ≠ 0) before one can conclude that B = C. If det(A) = 0, then B might not equal C, because the matrix equation AX = B will not have a unique solution for a non-invertible matrix A.
Matrix chain multiplication (or the matrix chain ordering problem[1]) is an optimization problem concerning the most efficient way to multiply a given sequence of matrices. The problem is not actually to perform the multiplications, but merely to decide the sequence of the matrix multiplications involved. The problem may be solved using dynamic ...
Freivalds' algorithm (named after Rūsiņš Mārtiņš Freivalds) is a probabilistic randomized algorithm used to verify matrix multiplication. Given three n × n matrices , , and , a general problem is to verify whether . A naïve algorithm would compute the product explicitly and compare term by term whether this product equals .