= Matrix element
{title2=$A_{ij}$}
A <matrix element> $A_{ij}$ is the entry in row $i$ and column $j$ of a <matrix> $A$.
For a <linear operator> $T$ on an <inner product space> with an <orthonormal basis> $(e_i)$, its <matrix elements> are $T_{ij}=\langle e_i,T e_j\rangle$, using the convention that the <inner product> is linear in its second argument. For a finite-dimensional space, the expansion $T e_j=\sum_iT_{ij}e_i$ explains why these numbers represent the <linear operator> in that <basis>. In <quantum mechanics>, the expression $\langle\phi|T|\psi\rangle$ is also called a <matrix element> between the two states; it need not be diagonal or an <expectation value>.
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