Let $T\colon V \to V$ be a linear operator on the vector space $V$ with characteristic polynomial $\lambda^4(\lambda-4)^5$ and minimal polynomial $\lambda(\lambda-4)$, then what is the rank of $T$?


closed as off-topic by Xander Henderson, Don Thousand, ArsenBerk, Toby Mak, I am Back Oct 20 '18 at 8:20

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  • $\begingroup$ So, what is your 'problem'? $\endgroup$ – GAVD Dec 3 '15 at 8:04
  • $\begingroup$ Have you tried anything? $\endgroup$ – ASKASK Dec 3 '15 at 8:07
  • $\begingroup$ Yes but not worked. $\endgroup$ – sharafat salam Dec 3 '15 at 8:16
  • $\begingroup$ Use Jordon form $\endgroup$ – Black-horse Dec 3 '15 at 9:14
  • $\begingroup$ Can you do it black horse? $\endgroup$ – sharafat salam Dec 3 '15 at 9:26

The Jordon form of that matrix will be $$ \begin{bmatrix} (0)_4&0\\ 0&(4)_5\\ \end{bmatrix} $$ where $(0)_4$ is $4\times 4$ zero matrix and $(4)_5$ is $5\times 5$ diagonal matrix with diagonal entries $4$. Hence the rank is $5$


The minimal polynomial is of the form $(x-a)(x-b)$ hence the matrix is diagonalizable and hence rank is 5


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