15 Diagonalization of symmetric matrices

For symmetric matrices the same observations apply as for nonsymmetric matrices in the previous section. But there are some further considerations.

Most importantly, the eigenvectors, if found using the class procedure, are orthonormal. So you can consider the eigenvectors $\vec e_1, \vec e_2, \ldots$ to be a rotated Cartesian basis. To make this clearer to other people, you should rename $\vec e_1, \vec e_2, \ldots$ to $\hat\imath ',\hat\jmath ',\ldots$.

The most important other thing to remember is that the transformation matrix

$$E=(\hat\imath ' \hat\jmath ' \ldots)$$

is now orthonormal. So be sure to use section 9 to find its inverse.

Note further that if the determinant of the transformation matrix $E$ is negative, the rotated coordinate system is also left-handed instead of right-handed. It corresponds to a mirroring of a coordinate besides the rotation. If this bothers you, multiply one of the eigenvectors by minus one.