Matrix unit explained

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 _{ij} . For example, the 3 by 3 matrix unit with i = 1 and j = 2 isA 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.

Properties

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 x n satisfies the relation$$E\_E\_\; =\; \backslash delta\_E\_,$$where \delta _{jk} 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]

E _{23} A = \left[ \begin{matrix} 0 & 0 & 0 \ a _{31} & a _{32} & a _{33} \ 0 & 0 & 0 \end{matrix}\right]. AE _{23} = \left[ \begin{matrix} 0 & 0 & a _{12} \ 0 & 0 & a _{22} \ 0 & 0 & a _{32} \end{matrix}\right].

Notes and References

1. Book: Artin, Michael . Algebra . Prentice Hall. 9.
2. Book: Lam, Tsit-Yuen . Tsit-Yuen Lam

   . Chapter 17: Matrix Rings . Lectures on Modules and Rings . Tsit-Yuen Lam . . 189 . . 1999 . 461–479.

3. Marcel Blattner . B-Rank: A top N Recommendation Algorithm . 2009 . physics.data-an . 0908.2741 .