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In mathematics, specifically 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.
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 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:
For the case of column vector c and row vector r, each with m components, the formula allows quick calculation of the determinant of a matrix that differs from the identity matrix by a matrix of rank 1: (+) = +. More generally, [14] for any invertible m × m matrix X,
the matrix exponential reduces to a plain product of the exponentials of the two respective pieces. This is a formula often used in physics, as it amounts to the analog of Euler's formula for Pauli spin matrices, that is rotations of the doublet representation of the group SU(2).
Matrix multiplication is defined in such a way that the product of two matrices is the matrix of the composition of the corresponding linear maps, and the product of a matrix and a column matrix is the column matrix representing the result of applying the represented linear map to the represented vector. It follows that the theory of finite ...
where is the matrix formed by replacing the i-th column of A by the column vector b. A more general version of Cramer's rule [13] considers the matrix equation = where the n × n matrix A has a nonzero determinant, and X, B are n × m matrices.
which can be found by stacking into matrix form a set of equations consisting of the above difference equation and the k – 1 equations =, …, + = +, giving a k-dimensional system of the first order in the stacked variable vector [+] in terms of its once-lagged value, and taking the characteristic equation of this system's matrix. This ...
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