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.[1]

• Step 1: find the symmetric matrix A which represents q and find its characteristic polynomial ${\displaystyle \Delta (t).}$
• Step 2: find the eigenvalues of A which are the roots of ${\displaystyle \Delta (t)}$.
• Step 3: for each eigenvalues ${\displaystyle \lambda }$ 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 ${\displaystyle P^{T}AP}$ will be the eigenvalues ${\displaystyle \lambda _{1},\dots ,\lambda _{n}}$ which correspond to the columns of P.

References

1. ^ Lipschutz, Seymour. 3000 Solved Problems in Linear Algebra.