Trace

Algebra Level 3

True or False?

If A A is an m × n m\times n matrix and B B is an n × m n\times m matrix, then tr ⁡ ( A B ) = tr ⁡ ( B A ) . \operatorname{tr}(AB)=\operatorname{tr}(BA).

True False

Brian Lie by Brian Lie , 26, China

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1 solution

Patrick Corn
Jan 3, 2019

Writing A = ( A i j ) A = (A_{ij}) and B = ( B k l ) , B = (B_{kl}), it's not hard to show that t r ( A B ) = ∑ i = 1 m ∑ j = 1 n A i j B j i t r ( B A ) = ∑ j = 1 n ∑ i = 1 m B j i A i j \begin{aligned} {\rm tr}(AB) &= \sum_{i=1}^m \sum_{j=1}^n A_{ij} B_{ji} \\ {\rm tr}(BA) &= \sum_{j=1}^n \sum_{i=1}^m B_{ji} A_{ij} \end{aligned} so they are equal.

0 Helpful 0 Interesting 2 Brilliant 0 Confused

I mean you don't even need to show it, it is just a knows fact that traces are commutative.

Ani B - 2 years, 5 months ago

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