Orthogonal diagonalization
From Wikipedia, the free encyclopedia
Template:Short description 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(Template:Itco).
- Step 2: Find the eigenvalues of A, which are the roots of Δ(Template:Itco).
- 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 PTATemplate:Px2P will be the eigenvalues λ1, ..., λn that correspond to the columns of P.
Such decomposition exists by the spectral theorem.
References
Page Template:Reflist/styles.css has no content.
- ^ Page Module:Citation/CS1/styles.css has no content.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
