Matrix unit

In mathematics, a matrix unit is the matrix with exactly one entry equal to 1, all other entries are equal to 0.

If A[i,j] denotes the matrix unit which is a square matrix of dimension n with 1 at the (i,j) entry, then A[i,j] A[k,l]=A[i,l] if j=k and is 0 otherwise.

Another notation for (r, c) is A_{r c}, following the convention for naming a single entry of a matrix. (Different letters are used in the "A" position to refer to matrix units on a different base set.) The composition rule may be expressed using the Kronecker delta as

X_{r c} X_{s d} = δ_{c s} X_{r d}.

With these rules, (I × I) ∪ {0} is a semigroup with zero. Its construction is analogous to that for other important semigroups, such as rectangular bands and Rees matrix semigroups. It also arises as the trace of the unique D-class of the bicyclic semigroup, meaning that it summarises how composition for members of that class interacts with the structure of the semigroup's principal ideals.

A semigroup of matrix units is 0-simple, because any two nonzero elements generate the same two-sided ideal (the entire semigroup), and the semigroup is non-null. The elements (r, c) and (s, d) are D-related via

(r, c) R (r, d) L (s, d),

as any pairs are R-related if they have the same first coordinate and L-related if they have the same second coordinate. All H-classes are singletons. The idempotents are the "square" matrix units (a, a) for a in I, together with 0.