title: Orthogonal diagonalization
text: # 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 $R^n$ 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 eigenvalue $\lambda$ 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 $P^{T}AP$ will be the eigenvalues $\lambda _{1},\dots ,\lambda _{n}$ which correspond to the columns of P.
  
  