Orthogonal diagonalization
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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.[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 that represents q and find its characteristic polynomial Δ(t).
- Step 2: Find the eigenvalues of A, which are the roots of Δ(t).
- Step 3: For each eigenvalue λ 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 PTAP will be the eigenvalues λ1, ..., λn that correspond to the columns of P.
References
[edit | edit source]- ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- ^ 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
