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 R {\displaystyle \mathbb {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 Δ ( t ) . {\displaystyle \Delta (t).}
• Step 2: find the eigenvalues of A which are the roots of Δ ( t ) {\displaystyle \Delta (t)} .
• Step 3: for each eigenvalue λ {\displaystyle \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 {\displaystyle \mathbb {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 A P {\displaystyle P^{T}AP} will be the eigenvalues λ 1 , … , λ n {\displaystyle \lambda _{1},\dots ,\lambda _{n}} 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