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 $𝐀$ and $𝐁$ are real/complex $𝑛 \times 𝑛$ matrices, then $t r 𝐀 𝐁 = t r 𝐀 t r 𝐁$ .
If $𝐀$ is nilpotent, $t r 𝐀 = 0$ .
$t r 𝐀 𝐁 = t r 𝐁 𝐀$
$t r 𝐓 - 1 𝐀 𝐓 = t r 𝐀$ for invertible $𝐓$

#state/develop | #lang/en | #SemBr