The ** matrix elements** of an operator are the entries of the matrix representation of the operator.

Consider a linear map from a vector space to the same vector space, i.e. , where and the orthonormal basis states span . The matrix representation of the equation is

where and are the coefficients of the vectors and respectively.

The matrix elements of are given by

Since the orthonormal basis states span , we have . So,

Similarly, and so

Comparing eq33 with eq32a, . Therefore, are the matrix elements of with respect to the basis states of .