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