Understanding eigenvalues and eigenvectors I am trying to understand eigen values for my control systems class , I will try to explain it below if someone could be kind enough to point out my mistakes ?
Lets $A$ be a transformation matrix , If we apply this  transformation certain vector , which doesn't change its direction but rather just scale in length with that transformation and that scaling can be represented in $\lambda$ such $$Ax = \lambda x$$
where $\lambda$ is a scalar , and x is the eigenvector of transformation $A$ , if we say that $A$ has eigenvalues then $A-\lambda I$ should be a singlular matrix

That's correct. To be very precise, if $\lambda = \pm 1$, then the length of a corresponding eigenvector $x$ doesn't change. 
But note also that a same matrix can have different eigenvalues with different eigenvectors. For instance, look at
$$A = \begin{bmatrix} 2 & 0\\ 0 & 3 \end{bmatrix}$$
For this matrix, $(1,0)$ is an eigenvector for the eigenvalue $2$ and $(0,1)$ is an eigenvector for the eigenvalue $3$. 
