User:Caliburn/Derivative of Trace Function

Theorem

Let $\tr \colon \R^{n \times n} \to \R$ be the trace function.

Let $A \in \R^{n \times n}$.

Then $\tr$ is Fréchet differentiable at $A$ with derivative having:

$\map {\paren {\mathrm D \map \tr A} } H = \tr H$

for each $H \in \R^{n \times n}$.

Proof

Let $H \ne 0$, then:

\(\ds \frac {\norm {\map \tr {A + H} - \tr A - \tr H} } {\norm H}\)

\(=\)

\(\ds \frac {\norm {\tr A + \tr H - \tr A - \tr H} } {\norm H}\)

Trace of Sum of Matrices is Sum of Traces

\(\ds \)

\(=\)

\(\ds 0\)

In particular:

$\ds \lim_{H \to 0} \frac {\norm {\map \tr {A + H} - \tr A - \tr H} } {\norm H} = 0$

so $\tr$ is differentiable at $A$ with derivative $\map {\paren {\mathrm D \map \tr A }} H = \map \tr H$.