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7—Operators and Matrices 171

chapter on Fourier series, section5.3, but it didn’t appear under this name. You will become familiar
with this class of operators when you hit quantum mechanics. Then they are ubiquitous. The same
theorem as for symmetric operators applies here, that the eigenvalues are real and that the eigenvectors
are orthogonal.
Orthogonaloperators satisfy



O(~u),O(~v)



=


~u,~v



. The most familiar example is rotation.
When you rotate two vectors, their magnitudes and the angle between them do not change. That’s all
that this equation says — scalar products are preserved by the transformation.
Unitaryoperators are the complex analog of orthogonal ones:



U(~u),U(~v)



=


~u,~v



, but all
the scalars are complex and the scalar product is modified accordingly.
The next couple you don’t see as often. Idempotentmeans that if you take the square of the
operator, it equals the original operator.
Nilpotentmeans that if you take successive powers of the operator you eventually reach the
zero operator.
Self-adjointin a finite dimensional vector space is exactly the same thing as Hermitian. In
infinite dimensions it is not, and in quantum mechanics the important operators are the self-adjoint
ones. The issues involved are a bit technical. As an aside, in infinite dimensions you need one extra
hypothesis for unitary and orthogonal: that they are invertible.

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