Orthogonal diagonalization
Introduction In linear algebra, an '''orthogonal diagonalization''' of a symmetric matrix is a diagonalization by means of an orthogonal change of coordinates.The following is an orthogonal diagonalization algorithm that diagonalizes a quadratic form ''q''(''x'') on '''R'''''n'' by means of an orthogonal change of coordinates ''X'' = ''PY''. * Step 1: find the symmetric matrix A which represents q and find its characteristic polynomial * Step 2: find the eigenvalues of A which are the roots of . * Step 3: for each eigenvalue  of A from step 2, find an orthogonal basis of its eigenspace. * Step 4: normalize all eigenvectors in step 3 which then form an orthonormal basis of '''R'''''n''. * Step 5: let P be the matrix whose columns are the normalized eigenvectors in step 4.Then X=PY is the required orthogonal change of coordinates, and the diagonal entries of  will be the eigenvalues  which correspond to the columns of P.
References * Maxime Bôcher (with E.P.R. DuVal)(1907) ''Introduction to Higher Algebra'',  § 45 Reduction of a quadratic form to a sum of squares via HathiTrust