I'm feeling like this is a disproof situation yet I cant come up with a working counter-example to the statement:
Let $A\in M_n(\mathbb{C})$ be an anti-hermitian matrix (such that $A=-A^*$ then for all $\alpha\in\mathbb{R}^+$ : $Rank(A-\alpha I)=n$
I'm aware that because $A$ is anti-hermitian, than all eigenvalues are pure imaginary.
What bothers me is the thought that $(A-\alpha I)$ could have a column $j$ where $a_{j,j}=0$ which then drops the rank of the matrix, in the case where $\alpha = a_{j,j}$ which is a possible(?) situation.
Not sure where to take it from here, any tips/assistance could be really helpful.
Thanks!