Math 2.0

Linear and multilinear algebra; matrix theory Definition

Trace of a matrix

Builds on Square matrix

Definition

Let RR be a commutative ring and let A=(aij)A = (a_{ij}) be an n×nn \times n square matrix over RR. The trace of AA is tr⁡A=∑i=1naii=a11+a22+⋯+ann.\tr A = \sum_{i=1}^{n} a_{ii} = a_{11} + a_{22} + \cdots + a_{nn}.

Remarks

The trace of (1234)\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} is 55. The trace is linear, it is invariant under transpose, and tr⁡(AB)=tr⁡(BA)\tr(AB) = \tr(BA). Similar matrices have the same trace, so the trace of a linear map is well defined. Over an algebraically closed field, the trace is the sum of the eigenvalues, counted with multiplicity.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Trace_(linear_algebra), Definition
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