Matrix unit

In linear algebra, a matrix unit is a matrix with only one nonzero entry with value 1. The matrix unit with a 1 in the ith row and jth column is denoted as . For example, the 3 by 3 matrix unit with i = 1 and j = 2 is A vector unit is a standard unit vector. A single-entry matrix generalizes the matrix unit for matrices with only one nonzero entry of any value, not necessarily of value 1. The set of m by n matrix units is a basis of the space of m by n matrices. The product of two matrix units of the same square shape satisfies the relation where is the Kronecker delta. The group of scalar n-by-n matrices over a ring R is the centralizer of the subset of n-by-n matrix units in the set of n-by-n matrices over R. The matrix norm (induced by the same two vector norms) of a matrix unit is equal to 1. When multiplied by another matrix, it isolates a specific row or column in arbitrary position.