Suppose I have a $N \times N$ square matrix $A$. We know that for any matrix $A$, we have $N$ eigen values and corresponding eigen vectors. So a $2 \times 2$ matrix have two eigen vectors and corresponding eigen values. Now I am in a confusion. Its regarding identity matrices. For those matrices any vector can be eigen vector right? So there is a possibility of infinite eigen vectors. But what about eigen values? They are still $\lambda_1 =1 , \lambda_2 =1$ ,right?
EDIT : I am considering vectors $(x,y)$ and $(a.x,a.y)$ as same family. $(a.x,a.y)$ = $a \times (x,y)$, where a is a scalar.
Best Answer
If an $N\times N$ matrix $A$ has an eigenvalue $\lambda$, there are infinitely many vectors $v$ satisfying $Av = \lambda v$; that is, $A$ has infinitely many eigenvectors for the eigenvector $\lambda$. In fact,
$$E_{\lambda} := \{v \in \mathbb{R}^N \mid Av = \lambda v\}$$
is a subspace of $\mathbb{R}^N$.
This answers your first question, but let me point out an error in your thinking.
When we say $A$ has $N$ eigenvalues, we mean that the characteristic equation for $A$, $|\lambda I - A| = 0$, has $n$ zeroes; this is always true by the fundamental theorem of algebra, but some of the eigenvalues may be complex (e.g. a $2\times 2$ rotation matrix). Note, we count eigenvalues with multiplicity, so $\lambda$ could be repeated multiple times. We call the order of the zero $\lambda$ the algebraic multiplicity of $\lambda$.
For any eigenvalue $\lambda$, some say that $\lambda$ has $k$ corresponding eigenvectors if $\dim E_{\lambda} = k$ (this terminology is not often defined but is instead used in verbal communication). If $A$ has distinct eigenvalues $\lambda_1, \dots, \lambda_M$, one might say that $A$ has $\dim E_{\lambda_1} + \dots + \dim E_{\lambda_M}$ corresponding eigenvectors. The dimension of $E_{\lambda}$ is called the geometric multiplicity of $\lambda$.
We have the following result relating the two notions of multiplicity:
There are cases where the geometric multipicity of $\lambda$ is strictly less than the algebraic multiplicity, and therefore $A$ has less than $N$ corresponding eigenvectors. For example,
$$A = \left[\begin{matrix}1 & 1\\ 0 & 1\end{matrix}\right]$$
has a repeated eigenvalue of $1$ but only has a one-dimensional eigenspace.
To reconcile the difference between geometric multiplicity and algebraic multiplicity, one can consider generalised eigenvectors.