Commuting Matrices

Two matrices A and B which satisfy


under matrix multiplication are said to be commuting.

In general, matrix multiplication is not commutative. Furthermore, in general there is no matrix inverse A^(-1) even when A!=0. Finally, AB can be zero even without A=0 or B=0. And when AB=0, we may still have BA!=0, a simple example of which is provided by

A=[0 1; 0 0]
B=[1 0; 0 0],

for which



 BA=[0 1; 0 0]=A

(Taussky 1957).

See also


This entry contributed by Ronald M. Aarts

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Gantmacher, F. R. Ch. 8 in The Theory of Matrices, Vol. 1. Providence, RI: Amer. Math. Soc., 1998.Taussky, O. "Commutativity in Finite Matrices." Amer. Math. Monthly 64, 229-235, 1957.

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Commuting Matrices

Cite this as:

Aarts, Ronald M. "Commuting Matrices." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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