In linear algebra, a matrix unit is a matrix with only one nonzero entry with value 1.[1][2] The matrix unit with a 1 in the ith row and jth column is denoted as E i j {\displaystyle E_{ij}} . For example, the 3 by 3 matrix unit with i = 1 and j = 2 is
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.[2]
The product of two matrix units of the same square shape n × × --> n {\displaystyle n\times n} satisfies the relation
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.[2]
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. For example, for any 3-by-3 matrix A:[3]
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