Could use some hints on solving some questions.
- Let be a complex × matrix with rank , where > . Define $ = (^*)^{-1} ^*$ . Show that $H$ is a self-adjoint idempotent matrix.
What I know so far: I understand that self-adjoint means $H=H^*$ and idempotent means $HH = H$. I also know that $(A^*A)$ is hermitian. How do I proceed from here and what matrix definitions would help me?
EDIT: Managed to find idempotent matrix by multiplying the right hand side by itself. Still figuring out the hermitian.
- Let ∶ ⟶ be a linear operator on a finite-dimensional complex inner product space . Suppose that is a normal operator. If there is some complex scalar , some unit vector , and some positive real number such that $‖() − ‖ < $ , show that there is some eigenvalue of such that $ | − | < $
Tried so far: Using the definition of eigenvalues and eigenvectors, I've set $$ to be an eigenvalue and $v$ to be an eigenvector of $T$, then the statement holds true because both will be $0$. But I feel like I'm missing something else and I need some help pointing to it.