Matrix diagonalization is the process of taking a square matrix and converting it into a special type of matrix--a so-called diagonal
matrix--that shares the same fundamental properties of the underlying matrix.
Matrix diagonalization is equivalent to transforming the underlying system of equations
into a special set of coordinate axes in which the matrix takes this canonical form.
Diagonalizing a matrix is also equivalent to finding the
matrix's eigenvalues, which turn out to be precisely
the entries of the diagonalized matrix. Similarly, the
eigenvectors make up the new set of axes corresponding
to the diagonal matrix.