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 . 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.[2]

The product of two matrix units of the same square shape satisfies the relation

where is the Kronecker delta.[2]

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]


  1. ^ Artin, Michael. Algebra. Prentice Hall. p. 9.
  2. ^ a b c d Lam, Tsit-Yuen (1999). "Chapter 17: Matrix Rings". Lectures on Modules and Rings. Graduate Texts in Mathematics. Vol. 189. Springer Science+Business Media. pp. 461–479.
  3. ^ Marcel Blattner (2009). "B-Rank: A top N Recommendation Algorithm". arXiv:0908.2741 [].