Orthogonal diagonalization
- Orthogonal diagonalization
-
In linear algebra, an orthogonal diagonalization of a square 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 Rn by means of an orthogonal change of coordinates X = PY.[1]
- Step 1: find the symmetric matrix A which 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 eigenvalues λ 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 Rn.
- 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 PTAP will be the eigenvalues 
which correspond to the columns of P.
References
- ^ Lipschutz, Seymour. 3000 Solved Problems in Linear Algebra.
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