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I have a question that asks "Given that $A^4 = 0$, and $A$ is not the zero matrix, prove that $\lambda = 0$ for all eigenvalues of $A$".

Now I have some intuition about this that would allow me to answer this geometrically, which is that if after 4 transformations of $A$ a vector is sent to the nullspace for sure, then its eigenvalues must be characteristic of that behavior, which is that they're all $0$. Also, if it's diagonalizable, then we can say $A^4 = 0$, and $A = S^{-1}VS$, then $A^4 = S^{-1}V^4S$. The diagonals are 0's and thus it's all 0's for A since any matrix multiply by the zero matrix must be 0.

However, what should I do if $A$ is not diagonalizable? I'm not sure if my geometric interpretation would be strong enough of an argument. Thank you!

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  • $\begingroup$ The minimal polynomial divides $x^4$. The roots of this are.... $\endgroup$ – Adam Hughes Jun 7 '16 at 0:27
  • $\begingroup$ @AdamHughes I haven't yet learned about "minimal polynomials", are they characteristic polynomials? $\endgroup$ – OneRaynyDay Jun 7 '16 at 0:27
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    $\begingroup$ Basically, the minimal one is just the smallest degree polynomial for which the matrix vanishes. If you want to talk characteristic polynomials, that gets a bit trickier because you don't know the order of the matrix (i.e. is it $2\times 2, 3\times 3,$ et cetera $\endgroup$ – Adam Hughes Jun 7 '16 at 0:30
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Suppose there's some eigenvalue $\lambda \neq 0$ and let $v$ be an associated eigenvector of $\lambda$ (remember that by definition, $v\neq 0$).

What is $A^nv$? What if you take $n=4$?

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    $\begingroup$ $A^nv$ would be $\lambda^nv$, and thus $\lambda^4v$ in the case of $n=4$. If $A^4v = 0$, then $\lambda^4v$ must be 0 as well. Ah okay, the only value of $\lambda$ that can be possible is 0 then! Thank you very much :) $\endgroup$ – OneRaynyDay Jun 7 '16 at 0:30
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If $A^4 = 0$, then the minimal polynomial of $A$ divides $x^4$. The eigenvalues of $A$ are exactly the roots of the minimal polynomial. Thus the only eigenvalue can be $0$.

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Generally, a matrix, say $A$, called a nilpotent matrix when there is a positive integer number like $k$ such that $A^k=0$. The smallest such $k$ is usually called the degree of $A$. You may see some equivalencies for a nilpotent matrix here: https://en.wikipedia.org/wiki/Nilpotent_matrix One of them is that all the eigenvalues of a matrix is zero if and only if it is a nilpotent one.

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