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Mathematics-Online course: Linear Algebra - Matrices - Matrix Operations

Trace

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The sum of the diagonal elements of a $ n\times n$ matrix $ A$ is called trace of $ A$,

$\displaystyle \operatorname{trace}(A)=\sum_{i=1}^n a_{ii}\,. $

For arbitrary $ (n \times n)$-matrices $ A, B$ we have

$\displaystyle \operatorname{trace}(AB) = \operatorname{trace}(BA) .$

Hence, for a regular matrix $ T$ and an arbitrary matrix $ A$ it follows, that

$\displaystyle \operatorname{trace}(T^{-1}AT) = \operatorname{trace}(A),$

which means that the trace is invariant under change of basis.
  automatically generated 4/21/2005