Frobenius inner product

For complex $m\times n$ matrices, the Frobenius inner product isHere $A^{\ast }$ is the conjugate transpose, and this convention is conjugate-linear in the first argument. It is the ordinary Euclidean inner product on all matrix entries, so $⟨A,A{⟩}_F={\sum }_{i,j}|a_{ij}{|}^2$ is positive for every nonzero matrix. Its induced norm is the Frobenius norm.

On real symmetric matrices it becomes $\mathrm{tr}(AB)$. In particular, for a real symmetric matrix $A$ and a real vector $x$,Thus a quadratic form evaluated at $x$ is a linear test of $A$ against a rank-one positive semidefinite matrix. This connects the geometry of matrices with quadratic measurements of vectors.