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Let $f$ be a linear operator on $\mathbb{R}^4$ whose minimal polynomial is $x^3-x^2$. I wish to find a linear operator $f$ satisfying this. I know that the minimal polynomial must divide the characteristic polynomial and have the same linear factors, in which case the characreristic polynomial must be either $A^3(A-I)$ or $A^2(A-I)^2$. Where do I go from here though?

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Check for the rational canonical forms, there are two possible cases: $$ \begin{pmatrix} 0&&& \\ &0&0&0 \\ &1&0&0 \\&0&1&1 \end{pmatrix}, \ \ \begin{pmatrix} 1&&& \\ &0&0&0 \\ &1&0&0 \\&0&1&1 \end{pmatrix} $$ So, the matrix with minimal polynomial $x^3-x^2$ must be similar to either one of the above.

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  • $\begingroup$ So how do I then use that in order to find the relevant matrix? $\endgroup$ – Si.0788 Dec 6 '16 at 20:16
  • $\begingroup$ For any nonsingular matrix $P$, multiply $P$ on the left and $P^{-1}$ on the right to the matrices above. $\endgroup$ – Sungjin Kim Dec 6 '16 at 20:42

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