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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.
In characteristic 2 the latter equality turns into = {, …,} (¯) what therefore provides an opportunity to polynomial-time calculate the Hamiltonian cycle polynomial of any unitary (i.e. such that = where is the identity n×n-matrix), because each minor of such a matrix coincides with its algebraic complement: = (+ /) where ...
Laplace's expansion by minors for computing the determinant along a row, column or diagonal extends to the permanent by ignoring all signs. [9]For every , = =,,,where , is the entry of the ith row and the jth column of B, and , is the permanent of the submatrix obtained by removing the ith row and the jth column of B.
In mathematics, the Bareiss algorithm, named after Erwin Bareiss, is an algorithm to calculate the determinant or the echelon form of a matrix with integer entries using only integer arithmetic; any divisions that are performed are guaranteed to be exact (there is no remainder).
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.
For matrix-matrix exponentials, there is a distinction between the left exponential Y X and the right exponential X Y, because the multiplication operator for matrix-to-matrix is not commutative. Moreover, If X is normal and non-singular, then X Y and Y X have the same set of eigenvalues. If X is normal and non-singular, Y is normal, and XY ...
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In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar.It is often denoted , .The operation is a component-wise inner product of two matrices as though they are vectors, and satisfies the axioms for an inner product.