I have just started studying representation theory of finite groups in finite dimensional vector spaces. I have question about writing a representation as direct sum of subrepresentations.
Now assume we have a representation $$G\rightarrow GL(V)$$$$p:g\rightarrow p_g$$ and let $V=W_1\oplus W_2...\oplus W_r$ and $W_i$ is $p_g$ invariant for all $i,g$ and $W_i=span\{v_{i1},v_{i2}...,v_{id_i}\}$ ($dim(W_i)=d_i$). If we set this base, $B=\{v_{11},...,v_{1d_1},...,v_{i1}...v_{id_i},...,v_{r1}...,v_{rdr}\}$ for $V$ then the matrix form of $p_g$ becomes a block diagonal matrix, say $A_g=diag\{A_1,A_2,...,A_r\}$ and each $A_i$ is subrepresentation, $A_g=A_1\oplus A_2\oplus ...\oplus A_n$
So my question is can i do the reverse? I mean if i have a representation $A_g$ -i know i can write it's Jordan form wihch is block diagonal and they are isomorphic representations since they are similar- can i always find or construct such bases as in example to make each subspace $A_g$ invariant?
For example let $A_g=diag\{A_1,...A_r\}$ and $A_1$ is $d_1\times d_1$ matrix, pick $W_1=span\{(1_F,0,...,0),(0,1_F,0,...,0),...(0,0,...,1_F\text{($d_1-th$ entry)},0,...,0)\}$ would not this representation be $W_i$ invariant? If so this should mean every finite dimensional representation which is not in form of a jordan block can be written as direct some of subrepresentations where those are the jordan blocks, right?
Thanks.