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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 $$\Delta (t)$$ .
Step 2: find the eigenvalues of A which are the roots of $$\Delta (t)$$ .
Step 3: for each eigenvalues $$\lambda$$ 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 $$P^T AP$$ will be the eigenvalues$$\lambda_{1} ,\dots ,\lambda_{n}$$ which correspond to the columns of P.

References

Lipschutz, Seymour. 3000 Solved Problems in Linear Algebra.