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Let $M$ be an $n \times n$ matrix over the field of complex numbers. Additionally assume that $M$ is invertible. Now let $E$ be the set of eigenvalues that is

$$E = \{\lambda \in \mathbb{C}: \exists v \in \mathbb{C}^n\setminus\{0\}, Mv=\lambda v\}$$

Now is it true, that $E\cap\{0\}^c \neq \emptyset$, i.e. does $M$ always have a non zero eigenvalue.

My thought are that by the fundamental theorem of algebra, I know that every complex polynomial, i.e. every polynomial with complex coefficients has at least one solution, and thus I can conclude that $E\neq \emptyset$. Now the general statement is clearly false if we don't assume that $M$ is invertible, since the zero matrix has only zero as an eigenvalue. So is the condition of invertibility sufficient?

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    $\begingroup$ Yes it must have. It is a direct consequence of the Jordan Normal Form. $\endgroup$
    – mathreadler
    Commented Aug 2, 2015 at 18:16
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    $\begingroup$ As the answers show, even more is true: every square complex matrix has a nonzero eigenvector. $\endgroup$
    – Lee Mosher
    Commented Aug 2, 2015 at 18:29
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    $\begingroup$ Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level. $\endgroup$
    – Fly by Night
    Commented Aug 2, 2015 at 19:13
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    $\begingroup$ @mathreadler the zero matrix has every eigenvector you can think of ... $\endgroup$
    – Gabriel Romon
    Commented Aug 2, 2015 at 19:20
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    $\begingroup$ What would non-zero eigenvector mean except "eigenvector associated with non-zero eigenvalue"? $\endgroup$
    – mathreadler
    Commented Aug 2, 2015 at 19:44

3 Answers 3

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Since every polynomial has a root over $\mathbb{C}$, the characteristic polynomial of any complex matrix must have a root, say $\lambda$. Then $\lambda$ is an eigenvalue of the matrix at hand. Since the matrix is assumed to be invertible, we have $\lambda \neq 0$.

Regarding the last statement, if $M$ has $0$ as eigenvalue, there is some non-zero eigenvector $x$: $Mx=0$, and $M$ is not invertible.

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  • $\begingroup$ the question was wether every matrix has a non-zero eigenvalue, not wether every matrix has a non-zero eigenvector, this answer is irrelevant. $\endgroup$
    – john
    Commented May 2, 2020 at 8:41
  • $\begingroup$ @john eigenvalues are roots of the characteristic polynomial, and roots of the characteristic polynomial are eigenvalues... $\endgroup$
    – Gabriel Romon
    Commented May 2, 2020 at 8:44
  • $\begingroup$ please read the question, its asking for "non-zero" eigenvalues, not just eigenvalues $\endgroup$
    – john
    Commented May 2, 2020 at 8:49
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    $\begingroup$ @john my answer was indeed flawed, thanks for pointing it out. $\endgroup$
    – Gabriel Romon
    Commented May 2, 2020 at 8:53
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Since $\mathbb{C}$ is algebraically closed, any matrix $A$ can be written (by change of basis) in Jordan normal form. Each Jordan block has exactly one eigenvector associated with it, so there is at least one eigenvector. (Worst case scenario, there is only one block; for example, this is the case with the matrices $\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$ and $\begin{bmatrix} i & 1 & 0 \\ 0 & i & 1 \\ 0 & 0 & i \end{bmatrix}$. In this case there is exactly one eigenvector.) We don't need to assume that $A$ is invertible.

For other possibly more elementary proofs, see here and here. Note that having at least one eigenvalue is equivalent to having at least one eigenvector.

Also note, that eigenvectors are by definition nonzero, so "nonzero eigenvector" is weird. Perhaps you mean nonzero eigenvalue.

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Let $A\in \Bbb C^{n\times n}$ be invertible $\iff |A|≠0$. Also $|A|=\lambda_1...\lambda_n$, therefore...

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