Orthogonal diagonalization

In linear algebra, an orthogonal diagonalization of a normal matrix (e.g. 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 that represents q and find its characteristic polynomial Δ(t).
• Step 2: Find the eigenvalues of A, which are the roots of Δ(t).
• Step 3: For each eigenvalue λ 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 P^TA‍P will be the eigenvalues λ₁, ..., λ_n that correspond to the columns of P.

Such decomposition exists by the spectral theorem.