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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.[1]

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.


Lipschutz, Seymour. 3000 Solved Problems in Linear Algebra.

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