Matries inequality with norms

Question

Let $P$ and $C \neq0$ a $q \times q$ matrices. I want to prove that there exists a positive constants $\alpha$ such under some assumptions under $P$ we have the inequality $${\left\| {P\left( {I - C} \right)x} \right\|_{{{\mathbb{R}}^q}}} \leqslant \alpha {\left\| {PCx} \right\|_{{{\mathbb{R}}^q}}}$$ for all $x$ in $\mathbb{R}$ whith $I$ is the identity matrix.

Thank you.

Answer 1

If $P=I$ and $C=0$, then there is no such positive constant, so that the inequality holds.