The following is an orthogonal diagonalization algorithm that diagonalizes a quadratic form q(x) on Rn 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 eigenvalues of A in step 2, find an orthogonal basis of its eigenspace.
- Step 4: normalize all eigenvectors in step 3 which then form an orthonormal basis of Rn.
- Step 5: let P be the matrix whose columns are the normalized eigenvectors in step 4.
The 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.
- 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