Orthogonal diagonalization

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 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 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 Rn. 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.