matrix unit (original) (raw)

A matrix unit is a matrix (over some ring with 1) whose entries are all 0 except in one cell, where it is 1.

For example, among the 3×2 matrices,

are the matrix units.

Let A and B be m×n and p×q matrices over R, and Ui⁢j an n×p matrix unit (over R). Then

1. 1. A⁢Ui⁢j is the m×p matrix whose jth column is the ith column of A, and 0 everywhere else, and
2. 1. Ui⁢j⁢B is the n×q matrix whose ith row is the jth row of B and 0 everywhere else.

Remarks. Let M=Mm×n⁢(R) be the set of all m by n matrices with entries in a ring R (with 1). Denote Ui⁢j the matrix unit in M whose cell (i,j) is 1.

• •
  M is a (left or right) R-module generated by the m×n matrix units.
• •
  When m=n, M has the structure of an algebra over R. The matrix units have the following properties:
  1. (a)
     Ui⁢j⁢Uk⁢ℓ=δj⁢k⁢Ui⁢ℓ, and
  2. (b)
     U11+⋯+Un⁢n=In,
     where δi⁢j is the Kronecker delta and In is the identity matrix. Note that the Ui⁢i form a complete set of pairwise orthogonal idempotents, meaning Ui⁢i⁢Ui⁢i=Ui⁢i and Ui⁢i⁢Uj⁢j=0 if i≠j.
• •
  In general, in a matrix ring S (consisting of, say, all n×n matrices), any set of n matrices satisfying the two properties above is called a full set of matrix units of S.
• •
  For example, if {Ui⁢j∣1≤i,j≤2} is the set of 2×2 matrix units over ℝ, then for any invertible matrix T, {T⁢Ui⁢j⁢T-1∣1≤i,j≤2} is a full set of matrix units.
• •
  If we embed R as a subring of Mn⁢(R), then R is the centralizer of the matrix units of Mn⁢(R), meaning that the only elements in Mn⁢(R) that commute with the matrix units are the elements in R.

References

• 1 T. Y. Lam, Lectures on Modules and Rings, Springer, New York, 1998.