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Matrix multiplication shares some properties with usual multiplication. However, matrix multiplication is not defined if the number of columns of the first factor differs from the number of rows of the second factor, and it is non-commutative, [10] even when the product remains defined after changing the order of the factors. [11] [12]
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:
An n × n matrix commutes with every other n × n matrix if and only if it is a scalar matrix, that is, a matrix of the form , where is the n × n identity matrix and is a scalar. In other words, the center of the group of n × n matrices under multiplication is the subgroup of scalar matrices.
In 1998, with his student Xiaohan Huang, Pan showed that matrix multiplication algorithms can take advantage of rectangular matrices with unbalanced aspect ratios, multiplying them more quickly than the time bounds one would obtain using square matrix multiplication algorithms.
Others, such as matrix addition, scalar multiplication, matrix multiplication, and row operations involve operations on matrix entries and therefore require that matrix entries are numbers or belong to a field or a ring. [8] In this section, it is supposed that matrix entries belong to a fixed ring, which is typically a field of numbers.
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 mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exponential of a matrix , which is involved in the closed-form solution of systems of linear differential equations .
The straightforward multiplication of a matrix that is X × Y by a matrix that is Y × Z requires XYZ ordinary multiplications and X(Y − 1)Z ordinary additions. In this context, it is typical to use the number of ordinary multiplications as a measure of the runtime complexity. If A is a 10 × 30 matrix, B is a 30 × 5 matrix, and C is a 5 × ...