Unitary transformations are like orthogonal transformations,
except we’re working with a complex inner product space. We’ll
focus on just the transformations that are unitary with respect
to the inner product itself. That is, we ask that and so we must
have just as we wrote for orthogonal transformations, but we
have to use the adjoint that’s appropriate to our complex inner
product. In this way, unitary and orthogonal transformations are
related in a way similar to that in which Hermitian and
symmetric forms are related. Now, we’ve got this running analogy
between endomorphisms on an inner product space and complex
numbers. Taking the adjoint is like complex conjugation, so
Hermitian transformations are like real numbers because they’re
equal to their own adjoints. But here, we’re looking at
transformations whose inverses are equal to their adjoints. What
does this look like in terms of our analogy? Well, we’ve noted
that a transformation composed with its own adjoint is a sure
way to get a positive-definite