# Bounding the difference in spectral norm of products of “close” matrices

For two collections of unitary matrices $U_1, \dots, U_n$ and $V_1, \dots, V_n$, satisfying $||U_i - V_i|| \leq \epsilon$ for all $i$, we have the following bound on the spectral norm of the difference of their products $$||U_1\dots U_n - V_1 \dots V_n || \leq n \epsilon.$$

I would like to find two generalizations of this relationship.

1) For two collections of positive semidefinite matrices $A_1, \dots, A_n$ and $B_1, \dots, B_n$ satisfying $||A_i - B_i|| \leq \epsilon$ for all $i$, what can we say about $$||A_1 \dots A_n - B_1 \dots B_n ||?$$

2) For two collections of positive semidefinite matrices $A_1, \dots, A_n$ and $B_1, \dots, B_n$ satisfying $|\text{Tr}(A_i-B_i)| \leq \epsilon$ for all $i$, what can we say about $$|\text{Tr}(A_1 \dots A_n - B_1 \dots B_n)|?$$