Orthogonal diagonalization

{{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.{{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}}

The following is an orthogonal diagonalization algorithm that diagonalizes a quadratic form q(x) on $\backslash mathbb\{R\}$ⁿ by means of an orthogonal change of coordinates X = PY.Seymour Lipschutz 3000 Solved Problems in Linear Algebra.

• Step 1: find the symmetric matrix A which represents q and find its characteristic polynomial $\backslash Delta(t).$
• Step 2: find the eigenvalues of A which are the roots of $\backslash Delta(t)$.
• Step 3: for each eigenvalue $\backslash lambda$ 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 $\backslash mathbb\{R\}$ⁿ.
• 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 $P^T\; A\; P$ will be the eigenvalues $\backslash lambda\_1,\backslash dots,\backslash lambda\_n$ which correspond to the columns of P.