Matrix trace pairing ($\phi (A,B)=\mathrm{tr}(AB)$)

On square matrices over a field, the trace of a product is a symmetric bilinear form. Pairing with the matrix unit $E_{ji}$ reads off entry $a_{ij}$, proving nondegeneracy. For real matrices of size at least two it is not positive-definite: a planar rotation block squares to $-I_2$ and has self-pairing $-2$. On real diagonalizable matrices its self-pairing is the sum of the squares of the real eigenvalues. This matrix pairing differs from the trace pairing of a field extension.