Linear Algebra, 5th Edition

6.5 Unitary and Orthogonal Operators and Their Matrices

In this section, we continue our analogy between complex numbers and linear operators. Recall that the adjoint of a linear operator acts similarly to the conjugate of a complex number (see, for example, Theorem 6.11 p. 357). A complex number z has length 1 if $z\overline{z}=1$. In this section, we study those linear operators T on an inner product space V such that $\text{TT*}=\text{T*T}=\text{I}$. We will see that these are precisely the linear operators that “preserve length” in the sense that $||\text{T}(x)||=||x||$ for all $x\in \text{V}$. As another characterization, we prove that, on a finite-dimensional complex inner product space, these are the normal operators whose eigenvalues all have absolute value 1.

In past chapters, we were interested ...