 Matrix element

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 $T_{ij}=⟨e_i,T{e}_j⟩$, using the convention that the inner product is linear in its second argument. For a finite-dimensional space, the expansion $T{e}_j={\sum }_i{T}_{ij}{e}_i$ explains why these numbers represent the linear operator in that basis. In quantum mechanics, the expression $⟨\varphi |T|\psi ⟩$ is also called a matrix element between the two states; it need not be diagonal or an expectation value.