# Uniqueness issue of direct sum decompostion of a representation?

1. Sep 20, 2011

### kof9595995

I'm having difficulty understanding this concept of uniqueness. What's the precise definition of it? Let say we have some direct sum decomposition,
(1)Are $\left( {\begin{array}{*{20}{c}} {{R_1}} & 0 \\ 0 & {{R_2}} \\ \end{array}} \right)$ and $\left( {\begin{array}{*{20}{c}} {{R_2}} & 0 \\ 0 & {{R_1}} \\ \end{array}} \right)$the same decomposition?
(2)Are $\left( {\begin{array}{*{20}{c}} {{R_1}} & 0 \\ 0 & {{R_2}} \\ \end{array}} \right)$ and$\left( {\begin{array}{*{20}{c}} {{R_1}} & 0 \\ 0 & {{U^{ - 1}}{R_2}U} \\ \end{array}} \right)$the same decomposition?