What's the "dimensionality" of a matrix (not dimension!)? I know that a $m\times n$ matrix has dimension... well, $m\times n$.
However, in a paper (not a mathematics paper!) I encountered the term dimensionality of a matrix, whose definition is nowhere to be found. 
Apparently, the dimensionality has to be a scalar, as it says e.g.:
"For all real, skew-symmetric matrices, the rank is 2 times the dimensionality, as eigenvalues come in conjugate imaginary pairs."
So, can anyone tell me what I have to deal with here?
Thanks!
 A: Trying to find information on the 'dimensionality' of a matrix is a bit difficult for me! But I think I figured it out in laymens terms:
The dimensionality of a vector space is the number of unique ways a set of vectors 'point' within the vector space. 
Consider a 2x2 vector space  $\vec{V} = \begin{bmatrix}
   a       & b \\
    c       & d \\
\end{bmatrix}$
The dimensionality of $\vec{V}$ is $4$, because it has four independant components that 'point' in different directions and $\vec{V}$ can be represented by a set of four basis vectors:
$A\begin{bmatrix} 1 & 0\\0 & 0\end{bmatrix}  + B\begin{bmatrix} 0 & 1\\0&0\end{bmatrix}+ C\begin{bmatrix} 0 & 0\\1&0\end{bmatrix} +D\begin{bmatrix} 0 & 0\\0&1\end{bmatrix}$. The basis vectors are arbitrary and all kinds of sets can satisfy the basis. But, you get what I mean.
Also consider the vector space $\vec{A}=A_x\hat{x} + A_y\hat{y} + 0\hat{z}$. It has dimensionality $2$ because the components only exist in the $x$-$y$ plane. 
Lastly, consider the vector space $\vec{A}=k\hat{x} + k\hat{y} + k\hat{z}$. This one is tricky, because $\vec{A}$ appears to point in three different directions. But since the components are all the same, the vector space is just a one dimensional line from $($-$\infty,$ -$\infty,$ -$\infty)$ to $(\infty, \infty, \infty)$. So the dimensionality is $1$.
Also, I am still confused a bit about vector space dimensionality, so if anyone cares to right my wrongs, please do! I'm really unclear about this kind of stuff and any help is appreciated. Thanks!
