Real symmetric matrices (or more generally, complex Hermitian matrices) always have real eigenvalues, and they are never defective. Their eigenvectors can, and in this class must, be taken orthonormal. (Mutually orthogonal and of length 1.)
For real symmetric matrices, initially find the eigenvectors like for a nonsymmetric matrix. However, after that, you must make the eigenvectors orthonormal in this class.
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The most general method to do that is using the Gram-Schmidt
procedure. This procedure works as follows. You start with the
nonorthogonal eigenvectors corresponding to the multiple eigenvalue,
You create your first good eigenvector as before, by
simply dividing by its length:
Now you need to make the remaining vectors in the set of
orthogonal to . You do that by removing from each vector
its vector component in the direction of , as follows:
Now you create your second good eigenvector by simply
dividing by its length
If there are still vectors left, you need to make each orthogonal
to . In the same way as before
Now you create your third good eigenvector by simply
dividing by its length
If there are still vectors left, you now have to make them orthogonal to . Then you can find by dividing by its length.
Continue like this for multiplicities greater than 4.
The general formula is
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