Orthogonal diagonalization without eigenvectors I stumbled onto a method for orthogonally diagonalizing a symmetric matrix with real entries and I was wondering what advantages (if any at all) it has over the eigenvector method.
It hinges on the fact that every symmetric matrix may be viewed as a dot product on some vector space. The idea is to use the Gram-Schmidt process to get zeros in every entry that is not on the diagonal.
What is this method called? What are its practical applications? (besides computing powers of matrices)
description of the method here since I cannot embed pictures directly into the post

It is a decomposition similar to the Cholesky called the $LDL$ decomposition
import numpy as np
from scipy.linalg import ldl

A = np.array([[3,-1,0],[-1,2,-1],[0,-1,3]])
A

array([[ 3, -1,  0],
       [-1,  2, -1],
       [ 0, -1,  3]])

lu, d, perm = ldl(A, lower =0)

lu

array([[ 1.        , -0.6       ,  0.        ],
       [ 0.        ,  1.        , -0.33333333],
       [ 0.        ,  0.        ,  1.        ]])

d


array([[2.4       , 0.        , 0.        ],
       [0.        , 1.66666667, 0.        ],
       [0.        , 0.        , 3.        ]]

Note that $2.4 = \frac{12}{5}$ and $1.\overline{666} = \frac{5}{3}$
