Linear algebra MOC

Trace

The trace $t r 𝑇$ of a linear endomorphism $𝑇 : 𝑉 \to 𝑉$ is the sum of its eigenvalues counted with (algebraic) multiplicity. #m/def/linalg This corresponds to the sum of main diagonal entries in any basis.

Properties

If $𝐀$ is nilpotent, $t r 𝐀 = 0$ .
The trace of a product is invariant under cyclic permutation of factors, e.g. $t r (𝐴 𝐵 𝐶) = t r (𝐵 𝐶 𝐴)$ .
$t r 𝐓 - 1 𝐀 𝐓 = t r 𝐀$ for invertible $𝐓$

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