Orthogonal diagonalization - Alchetron, the free social encyclopedia

In linear algebra, an orthogonal diagonalization of 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 Rⁿ 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 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 P T A P will be the eigenvalues λ 1 , … , λ n which correspond to the columns of P.

References

Orthogonal diagonalization Wikipedia

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