Orthogonal diagonalization

Orthogonal Diagonalization Algorithm

The following is the algorithm which 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 $\delta (t).$ Step 2: find the eigenvalues of A which are the roots of $\delta (t)$. Step 3: for each eigenvalues $\lambda$ 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 $R^{4}$. 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 P^T AP will be the eigenvalues Failed to parse (syntax error): {\displaystyle \lambda_1 ,…, \lambda_n} which correspond to the columns of P.