There are two equivalent mathematical ways of calculating physical properties, Schroedinger’s wave mechanics and Heisenberg’s matrix mechanics. In each systems are represented in terms of eigenstates and measurables as eigenvalues. In matrix mechanics the operator is represented by a Hermitian matrix of elements $$\langle m|\hat{Q} |n \rangle$$ which depends on the choice of basis set $$|m \rangle$$. Any state can represented by a normalised vector, which also depends on the basis set. The eigenvalues and eigenvectors of the matrix, however, do not depend on the choice of basis - the eigenvectors are, in fact, the eigenbasis of the operator.
For a set of basis ‘vectors’ of size N, there are $$N \times N$$ possible matrix elements.