{{short description|Method in linear algebra}} 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.<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 | access-date=12 November 2018 | page=411}}</ref>
The following is an orthogonal diagonalization algorithm that diagonalizes a quadratic form ''q''(''x'') on '''R'''<sup>''n''</sup> by means of an orthogonal change of coordinates ''X'' = ''PY''.<ref>Seymour Lipschutz ''3000 Solved Problems in Linear Algebra''.</ref> * Step 1: Find the symmetric matrix ''A'' that represents ''q'' and find its characteristic polynomial Δ({{itco|''t''}}). * Step 2: Find the eigenvalues of ''A'', which are the roots of Δ({{itco|''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'''<sup>''n''</sup>. * Step 5: Let ''P'' be the matrix whose columns are the normalized eigenvectors in step 4. Then {{nowrap|1=''X'' = ''PY''}} is the required orthogonal change of coordinates, and the diagonal entries of ''P''<sup>T</sup>''A''{{px2}}''P'' will be the eigenvalues ''λ''<sub>1</sub>, ..., ''λ''<sub>''n''</sub> that correspond to the columns of ''P''.
Such decomposition exists by the spectral theorem.
== References == {{reflist}} * Maxime Bôcher (with E.P.R. DuVal) (1907) ''Introduction to Higher Algebra'', [https://babel.hathitrust.org/cgi/pt?id=uc1.b4248862;view=1up;seq=147 § 45 Reduction of a quadratic form to a sum of squares] via HathiTrust
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Category:Linear algebra