Examples of inner products defined on a matrix vector space

Question

Is $\langle A,B \rangle = \operatorname{tr}(A^TB)$ the only inner product defined for $M(2\times 2)$? If there are others, what are some examples?

Answer 1

Let $\langle A,B \rangle = \operatorname{tr}(A^TB)$. For any invertible linear map $\Phi: M_{2 \times 2} \to M_{2 \times 2}$, the function $$ (A,B) = \langle \Phi(A),\Phi(B) \rangle $$ defines an inner product, and every inner product over $M_{2 \times 2}$ can be expressed in this fashion. For example: for any invertible $2 \times 2$ matrices $P,Q$, defining $\Phi(X) = PXQ$ leads to $$ (A,B) = \operatorname{tr}(Q^TA^TP^TPBQ) = \operatorname{tr}(A^T[P^TP]B[QQ^T]). $$ Note also that adding any inner products together yields another inner product.

Answer 2

Pick any basis $V_1,V_2,\ldots,V_{n^2}$ of $M_n(\mathbb R)$. Denote by $\langle X,Y\rangle_F$ the Frobenius inner product $\operatorname{tr}(X^TY)$. Then $$ f:A\mapsto\left(\langle A,V_1\rangle_F,\,\langle A,V_2\rangle_F,\,\ldots,\langle A,V_{n^2}\rangle_F\right) $$ is an isomorphism between $M_n(\mathbb R)$ and $\mathbb R^{n^2}$. Therefore we can define an inner product on $M_n(\mathbb R)$ by $$ \langle A,B\rangle:=\langle f(A),f(B)\rangle_{\mathbb R^{n^2}} =\sum_{j=1}^{n^2}\langle A,V_j\rangle_F\,\langle B,V_j\rangle_F,\tag{1} $$ where $\langle \cdot,\cdot\rangle_{\mathbb R^{n^2}}$ is the standard inner product (i.e. dot product) on $\mathbb R^{n^2}$. Actually the converse is also true: every inner product on $M_n(\mathbb R)$ can be expressed in the form of $(1)$ for some basis $V_1,V_2,\ldots,V_{n^2}$ of $M_n(\mathbb R)$.

For instance, for the Frobenius inner product on $M_2(\mathbb R)$, we can simply take $\{V_1,V_2,V_3,V_4\}$ as the standard basis $$ V_1=\pmatrix{1&0\\ 0&0}, V_2=\pmatrix{0&0\\ 1&0}, V_3=\pmatrix{0&1\\ 0&0}, V_4=\pmatrix{0&0\\ 0&1}. $$ One can directly verify that $\langle A,B\rangle_F$ is indeed equal to $\sum_{j=1}^4\langle A,V_j\rangle_F\,\langle B,V_j\rangle_F$.