Orthogonal diagonalization

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.

The X=PY is the required orthogonal change of coordinates, and the diagonal entries of ${\displaystyle P^{T}AP}$ will be the eigenvalues ${\displaystyle \lambda _{1},\dots ,\lambda _{n}}$ which correspond to the columns of P.

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