Does every invertible complex matrix have a non-zero eigenvalue? 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?
 A: 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.
A: 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.
A: Let $A\in \Bbb C^{n\times n}$ be invertible $\iff |A|≠0$. Also $|A|=\lambda_1...\lambda_n$, therefore...
