Is there a relation eigenvectors and unitary operator.

Question

I am trying the understand the spectral theorem as given in wikipedia link:

https://en.wikipedia.org/wiki/Spectral_theorem

I understand that eigenvectors are vectors and unitary operator is a matrix. My question is: for a positive definite matrix $A=UDU^H$ is the set of eigenvectors $U$ is unitary? In the above equation $D$ is a diagonal matrix of eigenvalues. Can we form a general relation between set of eigenvectors of an operator to unitary operator?

Answer 1

Look at the section on that page regarding "Normal Operators" (Normal operators include Self-Adjoint operators. There it is stated that "$A$ is normal if and only if there exists a unitary matrix $U$ such that $$A=U D U^*$$

So the spectral theorem directly addresses your question in that it is saying: You can diagonalize with a Unitary matrix precisely when the matrix is Normal.

Regarding positive definite matrices. If a matrix is positive-definite, then it is Self-Adjoint. You can find a proof of that in this MSE thread. So then, for a positive definite matrix, you can find a unitary matrix $U$ that diagonalizes it.

Proof Complex positive definite => self-adjoint

Answer 2

For a normal matrix $A$ (in particular, for a positive definite one), there exist a unitary matrix $U$ and diagonal matrix $D$ such that $A = U D U^H$. The diagonal elements of $D$ are the eigenvalues, and the columns of $U$ are eigenvectors of $A$ corresponding to those eigenvalues.