For some [[Matrix]] [[Vector Space]] $\mathcal M_{n \times n}(\R)$, its [[Inner Product]] is defined as: $\huge \braket{A, B} = \mathrm{Tr}(A^{\intercal}B) $ >[!example] >$\huge \begin{align} >\let A &= \mat{1&2\\3&4}\\ >\braket{A,A} &= \mathrm{Tr}\pa{ >\mat{1&3\\2&4} >\mat{1&2\\3&4} >}\\ >&= \mat{10 & 14 \\ 14 & 20} \\ >&= \boxed{30} >\end{align}$