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Trace

Linear algebra · Formula · Not yet linked from a lesson

Linear algebra

\operatorname{tr}(A) = \sum_{i=1}^{n} A_{ii}

The sum of a square matrix's diagonal. It equals the sum of the eigenvalues, survives a change of basis untouched, and lets you cycle matrices inside it — the trick behind many compact derivations of gradients.

Symbols

$A$: A square matrix.
$A_{ii}$: Its i-th diagonal entry.

Appears in

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Eigenvalue equation $A\mathbf{v} = \lambda \mathbf{v}, \quad \mathbf{v} \neq \mathbf{0}$
Matrix multiplication $(AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}$

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