Frobenius inner product ($⟨A,B{⟩}_F=\mathrm{tr}(A^{\ast }B)$)

For complex matrices, the inner product $⟨A,B{⟩}_F=\mathrm{tr}(A^{\ast }B)$ is conjugate-linear in its first argument. On real symmetric matrices it becomes $\mathrm{tr}(AB)$. In particular $⟨A,x{x}^T{⟩}_F=x^{T}Ax$, identifying quadratic forms with linear tests on matrices.