Orthogonal diagonalization
From HandWiki
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 [math]\displaystyle{ \Delta (t). }[/math]
- Step 2: find the eigenvalues of A which are the roots of [math]\displaystyle{ \Delta (t) }[/math].
- Step 3: for each eigenvalue [math]\displaystyle{ \lambda }[/math] 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 [math]\displaystyle{ P^T AP }[/math] will be the eigenvalues [math]\displaystyle{ \lambda_{1} ,\dots ,\lambda_{n} }[/math] which correspond to the columns of P.
References
- ↑ Poole, D. (2010) (in nl). Linear Algebra: A Modern Introduction. Cengage Learning. p. 411. ISBN 978-0-538-73545-2. https://books.google.com/books?id=FByELohRQd8C&pg=PA411. 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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