Orthogonal diagonalization: Difference between revisions
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In [[linear algebra]], an '''orthogonal diagonalization''' of a symmetric [[matrix (mathematics)|matrix]] is a [[diagonalizable matrix|diagonalization]] by means of an [[orthogonal matrix|orthogonal]] change of coordinates.<ref name="Poole 2010 p. 411">{{cite book | last=Poole | first=D. | title=Linear Algebra: A Modern Introduction | publisher=Cengage Learning | year=2010 | isbn=978-0-538-73545-2 | url=https://books.google.com/books?id=FByELohRQd8C&pg=PA411 | language=nl | access-date=12 November 2018 | page=411}}</ref> |
In [[linear algebra]], an '''orthogonal diagonalization''' of a symmetric [[matrix (mathematics)|matrix]] is a [[diagonalizable matrix|diagonalization]] by means of an [[orthogonal matrix|orthogonal]] change of coordinates.<ref name="Poole 2010 p. 411">{{cite book | last=Poole | first=D. | title=Linear Algebra: A Modern Introduction | publisher=Cengage Learning | year=2010 | isbn=978-0-538-73545-2 | url=https://books.google.com/books?id=FByELohRQd8C&pg=PA411 | language=nl | access-date=12 November 2018 | page=411}}</ref> |
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Revision as of 11:26, 12 November 2018
In linear algebra, an orthogonal diagonalization of 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 which represents q and find its characteristic polynomial
- Step 2: find the eigenvalues of A which are the roots of .
- Step 3: for each eigenvalues of A in 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.
The X=PY is the required orthogonal change of coordinates, and the diagonal entries of will be the eigenvalues which correspond to the columns of P.
References
- ^ Poole, D. (2010). Linear Algebra: A Modern Introduction (in Dutch). 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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