I thought that I knew what a unitary transform is, until I started thinking about it.
(2^n-ons are hypercomplex numbers that are related via the 2^n-on construction. Including n=3 the 2^n-on construction gives the same numbers as the Cayley-Dickson construction. From there the 2^n-ons are "nicer".)
I know the following:
· Unitary transformations are special kinds of normal transformations.
· In Hilbert space unitary transformations have eigenvectors that together form an orthogonal base of the Hilbert space.
· The eigenvalues are unit size numbers.
· They leave inner products of vectors untouched.
· The adjoint transformation equals the inverse of the original.
· Unitary transformations move vectors around in Hilbert space, but not their own eigenvectors.
· They move subspaces around in Hilbert space, but their eigenvectors stay put.
· A sequence of unitary transforms can move a subspace over a significant distance.
· A Fourier transform is a special kind of unitary transformation.
Now my questions:
1. Are the eigenvalues involved in the transfer of vectors?
2. If the Hilbert space is defined over the 2^n-ons, can the eigenvalues be 2^m-ons with m > n? (2^m-ons resemble 2^n-ons in their lower n dimensions)
3. What makes a unitary transform a Fourier transform?
4. Can the action of any unitary transform be represented as a trail of infinitesimal unitary transforms?
5. A subspace can be redefined by replacing the vectors that span this subspace. Can this redefinition be seen as (part of) a unitary transformation?
6. Can replacements be done freely? (Inertia learns that there exists a reaction in case of a disordered replacement! So, what is well ordered?)