I almost forgot to throw in this little observation about unitary and orthogonal matrices that will come in handy.
Let’s say we’ve got a unitary transformation and an orthonormal basis . We can write down the matrix as before
Now, each column is a vector. In particular, it’s the result of transforming a basis vector by .
What do these vectors have to do with each other? Well, let’s take their inner products and find out.
since preserves the inner product. That is the collection of columns of the matrix of form another orthonormal basis.
On the other hand, what if we have in mind some other orthonormal basis . We can write each of these vectors out in terms of the original basis
and even get a change-of-basis transformation (like we did for general linear transformations) defined by
so the are the matrix entries for with respect to the basis . This transformation will then be unitary.
Indeed, take arbitrary vectors and . Their inner product is
On the other hand, after acting by we find
since the basis is orthonormal as well.
To sum up: with respect to an orthonormal basis, the columns of a unitary matrix form another orthonormal basis. Conversely, writing any other orthonormal basis in terms of the original basis and using these coefficients as the columns of a matrix gives a unitary matrix. The same holds true for orthogonal matrices, with similar reasoning all the way through. And both of these are parallel to the situation for general linear transformations: the columns of an invertible matrix with respect to any basis form another basis, and conversely.