Sesquilinearform/Gramsche Matrix unter Basiswechsel/Fakt/Beweis

Beweis

Es ist

${}{\begin{aligned}\left\langle w_{r},w_{s}\right\rangle &=\left\langle \sum _{j=1}^{n}a_{rj}v_{j},\sum _{k=1}^{n}a_{sk}v_{k}\right\rangle \\&=\sum _{1\leq j,k\leq n}a_{rj}{\overline {a_{sk}}}\left\langle v_{j},v_{k}\right\rangle \\&=\sum _{1\leq j\leq n}a_{rj}{\left(\sum _{1\leq k\leq n}{\overline {a_{sk}}}\left\langle v_{j},v_{k}\right\rangle \right)}\\&={\left({A^{\text{tr}}}\circ {\left(G\circ {\overline {A}}\right)}\right)}_{rs}.\end{aligned}}$

Zur bewiesenen Aussage