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The polynomials in the diagonal of the Smith normal form or rational canonical form of a matrix are called its invariant factors.

A square matrix A such that A^2=I, where I is the identity matrix. An involutory matrix is its own matrix inverse.

Let A = [B D; E C] (1) A^(-1) = [W X; Y Z], (2) where B and W are k×k matrices. Then det(Z)det(A)=det(B). (3) The proof follows from equating determinants on the two sides of ...

Let S={x_1,...,x_n} be a set of n distinct positive integers. Then the matrix [S]_n having the least common multiple LCM(x_i,x_j) of x_i and x_j as its i,jth entry is called ...

A triangular matrix L of the form L_(ij)={a_(ij) for i>=j; 0 for i<j. (1) Written explicitly, L=[a_(11) 0 ... 0; a_(21) a_(22) ... 0; | | ... 0; a_(n1) a_(n2) ... a_(nn)]. ...

Denote the sum of two matrices A and B (of the same dimensions) by C=A+B. The sum is defined by adding entries with the same indices c_(ij)=a_(ij)+b_(ij) over all i and j. ...

A polynomial with matrix coefficients. An nth order matrix polynomial in a variable t is given by P(t)=A_0+A_1t+A_2t^2+...+A_nt^n, where A_k are p×p square matrices.

The natural norm induced by the L-infty-norm is called the maximum absolute row sum norm and is defined by ||A||_infty=max_(i)sum_(j=1)^n|a_(ij)| for a matrix A. This matrix ...

If two square n×n matrices A and B are simultaneously upper triangularizable by similarity transforms, then there is an ordering a_1, ..., a_n of the eigenvalues of A and ...

A matrix with 0 determinant whose determinant becomes nonzero when any element on or below the diagonal is changed from 0 to 1. An example is M=[1 -1 0 0; 0 0 -1 0; 1 1 1 -1; ...