Orthogonal diagonalization

In linear algebra, an orthogonal diagonalization of a symmetric matrix is a diagonalization by means of an orthogonal change of coordinates.[1]

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.[2]

  • Step 1: find the symmetric matrix A which represents q and find its characteristic polynomial
  • Step 2: find the eigenvalues of A which are the roots of .
  • Step 3: for each eigenvalues 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 will be the eigenvalues which correspond to the columns of P.


  1. Poole, D. (2010). Linear Algebra: A Modern Introduction (in Dutch). Cengage Learning. p. 411. ISBN 978-0-538-73545-2. Retrieved 12 November 2018.
  2. Seymour Lipschutz 3000 Solved Problems in Linear Algebra.

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