How to upper bound this quantity?

Question

Let $M,\tilde{M},P, \tilde{P}, Q$ and $\tilde{Q}$ be $n\times n$ non singular matrices. Assume $P$ and $\tilde{P}$ are orthogonal, and let $b$ be a vector in $\mathbb{R}^n$. Suppose that the following inequalities are given:

$\|\tilde{M}-M \|\le \epsilon _1$
$\|\tilde{P} -P \| \le \epsilon_2$ and
$\|\tilde{Q}-Q\|\le \epsilon_3$.

How can I use such information to upper bound the quantity
$\|\tilde{M}\tilde{P}\tilde{Q}\tilde{P}^Tb - MPQP^Tb\|$?

Answer 1

try to continue like this by inserting some creative zeros $$ \tilde M \tilde P \tilde Q \tilde P^T = (\tilde M-M)\tilde P \tilde Q \tilde P^T + M \tilde P \tilde Q \tilde P^T \\ = (\tilde M-M)\tilde P \tilde Q \tilde P^T + M (\tilde P -P)\tilde Q \tilde P^T + MP\tilde Q \tilde P^T\\ = \dots $$