nLab diagonal matrix

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Contents

Idea

A square matrix (A x,y)Mat n×n(R)(A_{x,y})\in Mat_{n \times n}(R) whose values for xyx\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 RR 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×n(R)Mat_{n \times n}(R).

References

See also

Last revised on May 21, 2024 at 10:02:28. See the history of this page for a list of all contributions to it.