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Given a $3\times 3$ matrix $N$ such that $N^3=0$, then which of the following are/is true?

  1. $N$ has a non zero eigenvector

  2. $N$ is similar to a diagonal matrix

  3. $N$ has $3$ linearly independent eigenvector

  4. $N$ is not similar to a diagonal matrix

Well, eigenvalues of $N$ are all zeroes and characteristic polynomial is $x^3=0$, clearly not diagonalizable. so only $1$ is true.

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    One of 2 and 4 has to be true....2012-06-13
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    I think you need to worry about if $N$ is the zero matrix or not.2012-06-13
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    But $N$ is nilpotent of degree 3, which (should) mean that $N^2 \neq 0$.2012-06-13
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    non-zero eigenvector is a pleonasm: eigenvectors are non-zero by definition2012-06-13
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    The body and title of the question give different information. Do we only have $N^3=0$, or do we also have that $N$ is nilpotent of degree $3$ (i.e. $N^2\ne0$)? If so, this should be added to the body of the question.2012-06-13
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    @MarcvanLeeuwen yes, simply "eigenvector" would suffice. And joriki's question needs to be settled to answer that, because the $0$-matrix doesn't have any. I believe the original questioner "meant" $N^3 = 0, N^2 \neq 0$, but may not be aware how crucial the second bit is.2012-06-13
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    I am sorry, $N$ was not nilpotent in the question.2012-06-13
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    @DavidWheeler Actually sometimes 2 is true and sometimes 4 is true, thus neither 2 nor 4 are true statements...2012-06-13
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    @DavidWheeler: Surely the $0$-matrix does have (non-zero) eigenvectors.2012-06-13

1 Answers 1

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"Clearly not diagonalizable" is not correct; if we know that $N^2\neq 0$, then you are correct (that would imply that the minimal polynomial of $N$ is also $x^3$, and since it is not square free then $N$ is not diagonalizable). But just from knowing that $N$ has characteristic polynomial $x^3$, we do not know whether $N$ is diagonalizable or not. It could be diagonalizable. Explicitly, $N$ is diagonalizable if and only if it is the zero matrix; prove it!

As noted, you cannot have 2 and 4 both false, since they are negations of each other and the excluded middle applies here. And 2 and 3 are logically equivalent for a $3\times 3$ matrix.

If the question explicitly states that $N^2\neq 0$, then you know that 2 is false. If the question does not explicitly state so, then you don't.