Orthogonal diagonalization

Last updated July 14, 2024

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.^[1]

The following is an orthogonal diagonalization algorithm that diagonalizes a quadratic form q(x) on $\mathbb {R}$ 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 $\Delta (t).$
Step 2: find the eigenvalues of A which are the roots of $\Delta (t)$ .
Step 3: for each eigenvalue $\lambda$ 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 $\mathbb {R}$ 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 $P^{T}AP$ will be the eigenvalues $\lambda _{1},\dots ,\lambda _{n}$ which correspond to the columns of P.

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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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[Poole_2010_p._411-1] 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.

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