## Unitary and Orthogonal Matrices and Orthonormal Bases

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.

[…] Now if we have an arbitrary orthonormal basis — say is a transformation on with the standard basis already floating around — we may want to work with the matrix of with respect to this basis. If this were our basis of eigenvectors, would have the diagonal matrix . But we may not be so lucky. Still, we can perform a change of basis using the basis of eigenvectors to fill in the columns of the change-of-basis matrix. And since we’re going from one orthonormal basis to another, this will be unitary! […]

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[…] already floating around for , we can use this new basis to perform a change of basis, which will be orthogonal (not unitary in this case). That is, we can write the matrix of any self-adjoint transformation as […]

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[…] numbers) we find that they’re orthonormal. But this means that the modified table is a unitary matrix, and thus its columns are orthonormal as well. We conclude […]

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