References: edX online course MIT 8.05 Week 6.

Hermitian and unitary operators are actually special cases of a more general type of operators known as *normal operators*. An operator is normal if it commutes with its adjoint:

A hermitian operator is normal because . A unitary operator is normal because and every operator commutes with its inverse.

**Theorem 1** * A normal operator remains normal after a similarity transformation with a unitary operator . *

*Proof:* Suppose we transform according to

Then

Thus is normal.

**Theorem 2** * If is an eigenvector of the normal operator with eigenvalue , it is also an eigenvector of the adjoint with eigenvalue .*

*Proof:* Suppose

Consider the vector

The norm of this vector satisfies

where in the third line we used 1 and in the last line we used 8. Since the norm of is 0, and since , from 9 we have

as required.

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