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.^[1]

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.^[2]

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.

References

^ Poole, D. (2010). Linear Algebra: A Modern Introduction. Cengage Learning. p. 411. ISBN 978-0-538-73545-2. Retrieved 12 November 2018.
^ Seymour Lipschutz 3000 Solved Problems in Linear Algebra.

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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[Poole_2010_p._411-1] Poole, D. (2010). Linear Algebra: A Modern Introduction. Cengage Learning. p. 411. ISBN 978-0-538-73545-2. Retrieved 12 November 2018.

[2] Seymour Lipschutz 3000 Solved Problems in Linear Algebra.

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