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Given a matrix equation Ax=b, the normal equation is that which minimizes the sum of the square differences between the left and right sides: A^(T)Ax=A^(T)b. It is called a ...

The characteristic polynomial is the polynomial left-hand side of the characteristic equation det(A-lambdaI)=0, (1) where A is a square matrix and I is the identity matrix of ...

Two square matrices A and B are called congruent if there exists a nonsingular matrix P such that B=P^(T)AP, where P^(T) is the transpose.

A statistic w on the symmetric group S_n is called a weighted inversion statistic if there exists an upper triangular matrix W=(w_(ij)) such that ...

Let the n×n matrix A satisfy the conditions of the Perron-Frobenius theorem and the n×n matrix C=c_(ij) satisfy |c_(ij)|<=a_(ij) for i,j=1, 2, ..., n. Then any eigenvalue ...

The Woodbury formula (A+UV^(T))^(-1)=A^(-1)-[A^(-1)U(I+V^(T)A^(-1)U)^(-1)V^(T)A^(-1)] is a formula that allows a perturbed matrix to be computed for a change to a given ...

The permanent of an n×n integer matrix with all entries either 0 or 1 is 0 iff the matrix contains an r×s submatrix of 0s with r+s=n+1. This result follows from the ...

The digraph intersection of two digraphs is the digraph whose vertex set (respectively, edge set) is the intersection of their vertex sets (respectively, edge sets). If n is ...

Suppose that A and B are two algebras and M is a unital A-B-bimodule. Then [A M; 0 B]={[a m; 0 b]:a in A,m in M,b in B} with the usual 2×2 matrix-like addition and ...

A procedure for decomposing an N×N matrix A into a product of a lower triangular matrix L and an upper triangular matrix U, LU=A. (1) LU decomposition is implemented in the ...