nLab diagonal matrix

Idea

A square matrix $(A_{x,y})\in Mat_{n \times n}(R)$ whose values for $x\neq y$ are all zero is called a diagonal matrix.

Examples

A diagonal matrix with value 1 on the diagonal is an identity matrix.

Properties

Schur lemma

Diagonal matrices always commute with all other square matrices of the same size (with respect to matrix multiplication). However, if $R$ is an algebraically closed field, then the converse is also true (Schur lemma): the subset of all diagonal matrices is precisely the center of the ring of matrices $Mat_{n \times n}(R)$.

Related concepts

• permutation matrix

• invertible matrix

• Smith normal form

References

See also

• Wikipedia, Diagonal matrices

• Wikipedia, Diagonalizable matrix