Chapter 6: Unitary and Orthogonal Operators

Properties of unitary and orthogonal operators:

If T is unitary (complex) or orthogonal (real), then:

1. T is norm-preserving: \|T(v)\|=\|v\|

2. T is inner product preserving: \langle T(u), T(v)\rangle=\langle u,v\rangle

3. T is invertible, and T^{-1}=T^*.

What are eigenvalues of unitary and orthogonal operators?

All eigenvalues lie on the unit circle in the complex plane. That is, if T is unitary or orthogonal and Tv=\lambda v, then |\lambda|=1.

What is a rotation matrix in \mathbb{R}², and why is it orthogonal?

R(\theta)=\left[\begin{array}{cc}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{array}\right]. This matrix preserves lengths and angles, and satisfies R^TR=I, hence is orthogonal.

Is reflection across a line through the origin an orthogonal transformation?

Yes. Reflections preserve distance and reverse orientation. The matrix of a reflection has determinant -1 but still satisfies A^TA=I.