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 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 Δ ( t ) . {\displaystyle \Delta (t).}
* Step 2: find the eigenvalues of A which are the roots of  Δ ( t ) {\displaystyle \Delta (t)} .
* Step 3: for each eigenvalue  λ {\displaystyle \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 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  P T A P {\displaystyle P^{T}AP} will be the eigenvalues  λ 1 , … , λ n {\displaystyle \lambda _{1},\dots ,\lambda _{n}} which correspond to the columns of P.

## References[edit]

1. ^ Poole, D. (2010). Linear Algebra: A Modern Introduction (in Dutch). Cengage Learning. p. 411\. ISBN 978-0-538-73545-2. Retrieved 12 November 2018.
2. ^ 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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