Eigenvector in Matrix Space? I am confused about this small thing that, while seemingly immaterial in problem solving, seems very strange:
An eigenvector is a vector in a space V such as if T is a linear transformation from V to V, there is a constant $\lambda$ such that: $T(v)=\lambda v$. This is easy to understand for any kind of "normal" vector space like $\mathbb{R}^n, \mathbb{C}^n$ etc. Howerver, in $M_{n \times n}^{\mathbb{R}}$, everything is represented by matrices.
So does one say that an eigen"vector" is a matrix X that fulfills: $T(X)=\lambda X$ ?  If so what is it called? Or is it just possible to use vectors like the other vector spaces to define an eigenvector/value?
(I apologize in advance for the silly question).
 A: If something is a Vector Space, meaning that it satisfies it's axioms, then the elements are called vectors. While we are dealing with matrices as a Vector Space, this may seem like a strange notion calling them "vectors".
However this is made up for by the generality of the wording in another way. Once something is shown to be a Vector Space, anything proved for vector spaces also hold for it.
As an example, take $M(\Bbb R)_{2\times 2}$, with the basis 
$\left(\begin{array}{cc} 
1 & 0   \\
0 & 0 
\end{array}\right)$, $\left(\begin{array}{cc} 
0 & 1   \\
0 & 0 
\end{array}\right)$, $\left(\begin{array}{cc} 
0 & 0   \\
1 & 0 
\end{array}\right)$, $\left(\begin{array}{cc} 
0 & 0   \\
0 & 1 
\end{array}\right)$.
Since any matrix is expressible as a linear combination of these, it becomes easier to notate them as* $(1,0,0,0),\ldots,(0,0,0,1)$, whereby now we are considering the 'vectors' (which are matrices of course) as their co-ordinates with respect to this basis. Or in more mathematical language, we're using the co-ordinate isomorphism to $\Bbb R^4$.
So for example $\left(\begin{array}{cc} 
a & b   \\
c & d 
\end{array}\right)$ becomes simply $(a,b,c,d)$.
Then linear transformations behave as you're used to them. You have four basis vectors, so the linear transformation will be a $4\times 4$ matrix in this basis.
*As an extra note, for the purposes of Matrix multiplcation, you probably want to write $(a,b,c,d)^T$ instead of $(a,b,c,d)$.
A: Definition:
If $V$ is a vector space and $F\colon V\to V$ is a linear map, then an eigenvector of $F$ is a vector $v\in V\setminus\{0\}$ with $F(v)=\lambda v$. 
In particular, if $V=\Bbb R^{m\times n}$, then $M\in V$ with $M\neq 0$ and $F(M)=\lambda M$ is called an eigenvector of $F$ and $\lambda$ its associated eigenvalue. Indeed, $M$ is a vector of $V$ (but this vector has the form of a matrix).
Another example, if $V=C([0,1])$, then $f\colon [0,1]\to\Bbb R$ with $f\neq 0$ and $F(f)(x)=\lambda f(x)$ for all $x\in[0,1]$ is also called an eigenvector of $F$ (sometimes a.k.a. eigenfunction).
