About: Orthogonal diagonalization

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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. Then 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.

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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 * Step 2: find the eigenvalues of A which are the roots of . * Step 3: for each eigenvalue of A from 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. Then 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. (en)
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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. Then 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. (en)
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  • Orthogonal diagonalization (en)
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