Commuting matrices ↩

aliases: []
shorthands: {}
created: 2021-10-22 00:02:42
modified: 2022-01-10 04:13:03

Or with the commutator:

A set of matrices commute if they commute pairwise.

Commuting and eigenspaces

Commuting matrices preserve each other's eigenspaces.

Commuting matrices are simultaneously triangularizable, i.e. there are bases over which they are both upper triangular. (There exists a similarity transformation that transforms all of them upper triangular.)

The converse is not necessarily true, i.e. if there exists a similarity transformation that makes them both upper triangular, then they are not necessarily commutative.

Simultaneously diagonalizable

If matrices and are simultaneously diagonalizable, i.e. if similarity matrix such that and are both diagonal, then and commute.

The converse is not necessarily true here as well trivially: if and commute, then they are not necessarily simultaneously diagonalizable, since one of them may not be diagonalizable.

Hermitian matrices and eigenspaces

Two Hermitian matrices commute if their eigenspaces coincide. (See Operators with a common set of eigenvectors commute)
In other words: two Hermitian matrices without multiple eigenvalues commute if they share the same set of eigenvectors. (Because eigenspaces are the span of eigenvectors.)

Other properties

• The property of commutation is not transitive: may commute with and as well, but still and do not commute.
• An matrix commutes with every other scalar matrix¹.

Examples

• The identity matrix commutes with all matrices
• Every diagonal matrix commutes with all other diagonal matrices
• If the product of two symmetric matrices is symmetric, then they must commute