Hello!
Any kind soul can provide me with a counter example for this:
False statement: Every square matrix has an eigenvalue.
Thanks
$\displaystyle Det|M-\lambda I|=0$
$\displaystyle \begin{vmatrix}
0-\lambda&0\\
0&0-\lambda
\end{vmatrix}=0$
$\displaystyle \lambda^2=0 \Rightarrow \ \lambda=0$
But when $\displaystyle \lambda=0$, we get an eigenvalue of $\displaystyle \begin{pmatrix}
0\\
0 \end{pmatrix}$.
The problem is that this 0 vector has no direction which contradicts the definition of an eigenvalue.
(ie. eigenvectors have the same direction when they undergo a transformation. They're only scaled by the eigenvalue. A 0 vector does not have a direction so cannot be an eigenvector).
An eigen vector $\displaystyle x$ of $\displaystyle A$ is a non-zero vector such that:
$\displaystyle Ax= \lambda x$
for some scalar $\displaystyle \lambda$ (there is usually no requirement that $\displaystyle \lambda$ be non-zero here). So every non-zero vector is an eigen vector of the zero matrix, and $\displaystyle 0$ is the corresponding eigenvalue.
So yes, I was wrong the zero matrix does have an eigen value (I had what had to be non-zero confused).
CB