  We draw attention to the fact that a Hermitian matrix is always
diagonalizable and has real discrete spectrum whereas the Hermitian
Schr{\"o}dinger Hamiltonian: $H=p^2/2\mu+V(x)$, may not be so. For instance
when $V(x)=x, x^3, -x^2$, $H$ does not have even one real discrete eigenvalue.
Textbooks do not highlight this distinction. However, if $H$ has real discrete
spectrum, by virtue of the expansion theorem, one can convert the eigenvalue
problem $H\psi_n=E_n \psi_n$ into a matrix and get eigenvalues $E_n$ by
diagonalizing the matrix. We show, that the thus obtained $E_n$ could be
accurate, provided $H$ is devoid of scattering states. We suggest that this
could be a simple and apt way to introduce the method of Linear Combination of
Atomic Orbitals (LCAO) for finding the spectra of molecules. In textbooks,
usually the method of matrix-diagonalization appears meagerly as a degenerate
perturbation theory for more than one dimensions.
