= Matrix trace pairing
{title2=$\varphi(A,B)=\operatorname{tr}(AB)$}
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.
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