and in the other way, orthogonal nullspace and range require symmetry?
Thanks,
R
and in the other way, orthogonal nullspace and range require symmetry?
Thanks,
R
Hint: any $v\in V$ can be written uniquely as $v_1+v_2$, where $v_1$ is in the nullspace of $A$ and $v_2$ in the range space. Thus written, $Av=v_2$.
Let $f^2=f$. Then $f$ is self-adjoint iff $f = f^* f$ iff $\langle fx,y \rangle=\langle f^* f,y \rangle $ iff $\langle fx,y \rangle =\langle fx,fy \rangle$ iff $\langle fx,fy-y \rangle =0$ for all $x,y$ iff $ran(f)$ and $ker(f)$ are orthogonal.