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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.

The following is an orthogonal diagonalization algorithm that diagonalizes a quadratic form q(x) on ${\displaystyle \mathbb {R} }$ by means of an orthogonal change of coordinates X = PY.

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

References

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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