Orthogonal diagonalization explained

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.^[1]

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.^[2] \Delta(t).

• Step 2: find the eigenvalues of A which are the roots of

\Delta(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

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 will be the eigenvalues λ _{1,...,λ} _{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

Notes and References

1. Book: Poole, D. . Linear Algebra: A Modern Introduction . Cengage Learning . 2010 . 978-0-538-73545-2 . nl . 12 November 2018 . 411.
2. [Seymour Lipschutz]