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$A$ is matrix under $R$ which I know the following information about it:

$f_A(x)=(x+2)^4x^4$- Characteristic polynomial

$m_A(x)=(x+2)^2x^2$- Minimal polynomial.

I'm trying to find out

(i) $A$'s rank

(ii) $\dim$ $\ker(A+2I)^2$

(iii) $\dim$ $\ker (A+2I)^4$

(iv) the characteristic polynomial of $B=A^2-4A+3I$.

I believe that I don't have enough information to determine none of the above.

By the power of $x$ in the minimal polynomial I know that the biggest Jordan block of eigenvalue 0 is of size 2, so there can be two options of Jordan form for this eigenvalue: $(J_2(0),J_2(0))$ or $(J_2(0),J_1(0),J_1(0))$, therefore $A$'s rank can be $2$ or $3$. I'm wrong, please correct me.

How can I compute the rest?

Thanks for the answers.

  • 0
    Aren't there two options for (i)? I still don't manage to understand if I'm wrong or right.2011-08-21

1 Answers 1

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If the Jordan form of A is C then let P be invertible such that $A=PCP^{-1}$ then $(A+2I)^2=P(C+2I)^2P^{-1}\;\rightarrow\; \dim\ker((A+2I)^2)=\dim\ker((C+2I)^2)$ and you know exactly how $(C+2I)^2$ looks like (well, at least the part of the kernel).

The same operation should help you solve the rest of the problems

  • 0
    For example, can you see the rank of the following Jordan block structure? (\lambda I - J) = \begin{pmatrix} 0&1&0\\0&0&1\\0 &0&0\end{pmatrix}. Compute the square of this and check the rank of the result. This will give you an idea of rank change and the structure of the Jordan block. Note that the rank of the matrix is related to the number of the zero eigenvalues.2011-08-21