Simple jordan canonical form question

[franky2727]

two part question to which i have answered the first part and am stuck on the 2nd part
find r and invertible real matrices Q and P such that Q^-1AP=(I_r,0),(0,0)
where each 0 denotes a matrix of zeroes(not necessarily the same size in each case)

second part being paying special attention to the order of the vectors, write down bases of R³ with respect to which Q^-1AP represents the mapping x|->Ax

answers to the first part are r=1 Q^-1=(100),(-210),(-301) and therefore Q=(100),(1/2,1,0),(1/3,0,1)
and P=(1-21),(010),(000)
can show the working for this if you want but don't think its required for the second part, however i have no idea where to start with the 2nd part and would apprechiate a shove in the right direction, thanks

 

[mighty2000]

For the second part, we need to find bases of R3 that represent the mapping x|->Ax.

First, let's consider the basis of R3 given by the standard basis vectors: e1=(1,0,0), e2=(0,1,0), e3=(0,0,1).

We can represent these basis vectors in terms of the new basis vectors by multiplying them with Q-1:
Q-1e1 = (1,1/2,1/3), Q-1e2 = (0,1,0), Q-1e3 = (0,0,1).

These vectors form a basis for R3 with respect to which Q-1AP represents the mapping x|->Ax.

Another way to find a basis is to consider the columns of Q-1AP. The first column (1,0) represents the mapping x|->Ax1, where x1 is the first column of x. Similarly, the second column (0,0) represents the mapping x|->Ax2, where x2 is the second column of x.

So, we can take the basis vectors for R3 to be the columns of Q-1AP:
(1,0,0) and (0,0,1).

Note that these two bases are not unique, there are infinitely many bases that can represent the mapping x|->Ax. These are just two examples.