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Because matrix multiplication is not commutative, one can also define a left division or so-called backslash-division as A \ B = A −1 B. For this to be well defined, B −1 need not exist, however A −1 does need to exist. To avoid confusion, division as defined by A / B = AB −1 is sometimes called right division or slash-division in this ...
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m×n matrix and B is an n×p matrix, then their matrix product AB is the m×p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column ...
The matrix A is invertible if and only if the left block can be reduced to the identity matrix I; in this case the right block of the final matrix is A −1. If the algorithm is unable to reduce the left block to I, then A is not invertible.
In the mathematical discipline of numerical linear algebra, a matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. Many iterative methods (for example, for systems of differential equations) depend upon the direct solution of matrix equations involving matrices more general than tridiagonal matrices.
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.
Another abbreviated method is polynomial short division (Blomqvist's method). Polynomial long division is an algorithm that implements the Euclidean division of polynomials, which starting from two polynomials A (the dividend) and B (the divisor) produces, if B is not zero, a quotient Q and a remainder R such that A = BQ + R,
Since the only invertible element is 1, division is the identity function. Elements of GF( p n ) may be represented as polynomials of degree strictly less than n over GF( p ). Operations are then performed modulo m(x) where m(x) is an irreducible polynomial of degree n over GF( p ), for instance using polynomial long division .