Diagonalizing a Matrix is equivalent to finding the Eigenvectors and Eigenvalues. The Eigenvalues make up the entries of the diagonalized Matrix, and the Eigenvectors make up the new set of axes corresponding to the Diagonal Matrix.
See also Diagonal Matrix, Eigenvalue, Eigenvector
Arfken, G. ``Diagonalization of Matrices.'' §4.6 in Mathematical Methods for Physicists, 3rd ed.
Orlando, FL: Academic Press, pp. 217-229, 1985.