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 $ℝ$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 $\mathrm{\Delta }(t).$
• Step 2: find the eigenvalues of A which are the roots of $\mathrm{\Delta }(t)$.
• 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.

References

• 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

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