When is matrix multiplication commutative?

I know that matrix multiplication in general is not commutative. So, in general:

A,BRn×n:ABBA

But for some matrices, this equations holds, e.g. A = Identity or A = Null-matrix BRn×n.

I think I remember that a group of special matrices (was it O(n), the group of orthogonal matrices?) exist, for which matrix multiplication is commutative.

For which matrices A,BRn×n is AB=BA?

Answer

Two matrices that are simultaneously diagonalizable are always commutative.

Proof: Let A, B be two such n×n matrices over a base field K, v1,,vn a basis of Eigenvectors for A. Since A and B are simultaneously diagonalizable, such a basis exists and is also a basis of Eigenvectors for B. Denote the corresponding Eigenvalues of A by λ1,λn and those of B by μ1,,μn.

Then it is known that there is a matrix T whose columns are v1,,vn such that T1AT=:DA and T1BT=:DB are diagonal matrices. Since DA and DB trivially commute (explicit calculation shows this), we have AB=TDAT1TDBT1=TDADBT1=TDBDAT1=TDBT1TDAT1=BA.

Attribution
Source : Link , Question Author : Martin Thoma , Answer Author : Community

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