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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

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