TOPICS
Search

Commuting Matrices


Two matrices A and B which satisfy

 AB=BA
(1)

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]
(2)
B=[1 0; 0 0],
(3)

for which

 AB=0,
(4)

but

 BA=[0 1; 0 0]=A
(5)

(Taussky 1957).


See also

Commutative

This entry contributed by Ronald M. Aarts

Explore with Wolfram|Alpha

References

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.

Referenced on Wolfram|Alpha

Commuting Matrices

Cite this as:

Aarts, Ronald M. "Commuting Matrices." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CommutingMatrices.html

Subject classifications