Consider a Krylov subspace $K_m=\mathrm{span}\{v,Pv,...,P^{m-1}v\}$, for $P$ a square matrix and a nonzero vector $v$. Let $H_m$ represent the projection of $P$ (seen as an application) restricted to $K_m$, onto $K_m$. That is, for $V_m=[v_1,...,v_m]$ a basis of $K_m$:

$ H_m = V^T_m P V_m $

Assume, if necessary, that $V_m$ is orthonormal ($V^T_m V_m =I$), and that for no $m'<m$ it holds that $K_{m'}=K_m$.

**Question**: is it true that all eigenvalues of $H_m$ (Ritz eigenvalues) have geometric multiplicity 1, and what is a (simple) proof of this fact?
Any reference will be appreciated.