# Orthogonal diagonalization

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 **R**^{n} 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
**R**^{n}. - 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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