Fine Art


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.

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

Retrieved from ""
All text is available under the terms of the GNU Free Documentation License

Home - Hellenica World