What Are the Possible Jordan Forms and Characteristic Polynomial of T?

[darksidemath]

Hi guys i have this problem in my linear algebra curse . let $T:\mathbb{Q}^3→\mathbb{Q}^3$ a linear application s.t $(T^7+2I)(T^2+3T+2I)^2=0$
can you find all possible Jordan forms of T and related characteristic polynomial ? I am totally lost and that is the first time i see this type of problem

 

[HOI]

First, do you understand what the "Jordan form" of a matrix is? It is another matrix having the real eigenvalues of the original matrix on its main diagonal possibly with "1" just above each diagonal. If all eigenvalues are distinct, the matrix is diagonalizable and its "Jordan form" IS that diagonal matrix. When there is a duplicate eigenvalue, there will be a block with that eigenvalue on the diagonal as many times as the multiplicity of the eigenvalue and "1" just above each eigenvalue.

Note that x^3+ 3x+ 2= (x+ 1)(x+ 2). So -1 and -2 are eigenvalues and the "Jordan form" will include the block \(\displaystyle \begin{bmatrix}-1 & 0 \\ 0 & -2\end{bmatrix}\).

The other factor, \(\displaystyle x^7+ 2\) has the single real zero, \(\displaystyle -\sqrt[7]{2}\) and 6 complex roots, three pairs of complex conjugates. The Jordan form matrix will have \(\displaystyle -\sqrt[7]{2}\) on the main diagonal. a pair of complex conjugates, say a+ bi and a- bi, will give a submatrix of the form \(\displaystyle \begin{bmatrix}a & b \\ -b & a \end{bmatrix}\)