The geometry meaning of Unitary matrix/operator

Question

After studying well about the unitary,self adjoint and normal matrices and operator, I can say that they have pretty intresting characteristics, but I do not know how to visualize them in low dimensions spaces, I have seen some youtube clips about visualize a linear transformation and matrix multiplication but it was for general case, I still can't imagine a visualize for those operators with all of their characteristics.

Answer 1

Long story short: there is no good way to completely "visualize" complex matrices with any useful generality. Even in the smallest non-trivial case, we are looking at a transformation over $\Bbb C^2$, which from a "geometric" standpoint is really a $4$-dimensional space.

With that said: with matrices and with other "complicated" mathematical objects, "visualization" in the usual sense is not always necessary to get a feeling for a mathematical object, and this includes complex matrices. As an analogy, I suggest you watch this video from 3Blue1Brown about 10-dimensional spheres and boxes, which are "visualized" (in a limited sense) in terms of "sliders". Note that there is really nothing geometric about a row of 10 sliders. Nevertheless, we can leverage our understanding of this representation to get a "feeling" for the fact that the volume of a box grows faster than the volume of the box's inscribed sphere as the number of dimensions is increased.

Similarly, here is a limited way in which normal matrices (which include unitary, self-adjoint, and skew-adjoint operators) can be visualized. When a real matrix is diagonalized with real eigenvalues, the picture associated with the diagonalization of a linear transformation is one of space being "stretched, squished, or flipped" along the directions corresponding to the eigenvectors of the transformation.

In the case where a complex matrix can be diagonalized with real eigenvalues (e.g. a self-adjoint operator), the span of a single vector in $\Bbb C^n$ is something that would normally be visualized as $2$-dimensional, so that the stretch/squish/flip occurs uniformly across the entirety of this "2-dimensional" complex line. With that established, we can say that the complex eigenvalue $\lambda = re^{i \theta}$ encodes an expansion by factor $r>0$ followed by a rotation by angle $\theta$ within this complex line. The spectral theorem tells us that this is enough to visualize any normal operator, and that for any normal operator, these eigenspaces will be mutually orthogonal.

With that, we can still understand the notion of independent directions and the action of the linear transformation along each of these direction, and often this is enough. What we lose, however, is our ability to visualize these directions at the same time.