Linear and multilinear algebra; matrix theory Matrices and matrix algebra Proposition

Trace of a product is symmetric

Builds on Trace of a matrix Matrix multiplication

Statement

Let $R$ be a commutative ring. Let $A$ be an $m \times n$ matrix and $B$ an $n \times m$ matrix over $R$ . Then $\tr(AB) = \tr(BA)$ , where $\tr$ is the trace.

Remarks

Both sides equal $\sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij} b_{ji}$ . The matrices $AB$ and $BA$ can have different sizes, and the identity still holds. A consequence is $\tr(P^{-1} A P) = \tr A$ , so similar matrices have the same trace. For three matrices, only cyclic permutations are allowed in general: $\tr(ABC) = \tr(BCA)$ , but $\tr(ABC)$ can differ from $\tr(BAC)$ .

Reading path

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Matrix definition
Matrix multiplication definition
Square matrix definition
Trace of a matrix definition
Trace of a product is symmetric statement

Tag	`0Y4NF1DRKD`
URI	`m2://0Y4NF1DRKD`
Kind	statement, proposition
Writing	W0 stub
MSC	15A15
Status	This copy of the site has no build data. Run `m2 status 0Y4NF1DRKD`.
Uses	Trace of a matrix, Matrix multiplication

Trace of a product is symmetric

Statement

Remarks

Reading path

Further reading

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