I am currently working on constructing an index to determine the degree of scattering of points in two dimensions relative to origo, for a visual psychological experiment. For the experiment I need a relative index of scattering that takes into account distance from the fixation point, at the center of the screen - origo. As the basis for computations, I have found the following computation appealing: I construct an Euclidian distance matrix of Pythagorean-based vectors, which is ordered by the proximity of points to origo. The matrix is ordered from smallest to largest distance to origo. Now, I would like to represent each combination of distances, that is each matrix, with a single index. For this I have considered the determinant, but I am not so certain what it would mean in the current case.
(Using graph theory, I have considered computing the average path length between the points and perhaps some other measure. However, these measures do not take into account the distance to origo. It would be possible, perhaps, to normalize these measures relative to the average distance to origo, but such a measure can be hard to interpret.)
For a matrix $$n\times n$$
where $n$ is the number of points ordered from smallest to largest distance from origo, what does the determinant represent?
Since you are not familiar with linear algebra, I'll try to elaborate an explanation through some examples: the first ones should be familiar, but I am going to generalize some results without going into hard proof details.
Let's start with a two dimensional vector space, namely $\mathbb{R}^2$. This is the set of two-dimensional vectors in the euclidean plane. What does the determinant of a matrix represent?
Consider the matrix $$A=\begin{bmatrix}1 & 2 \\0 & 1\end{bmatrix}$$ It's determinant is $\det(A)=1\cdot 1- 2\cdot 0=1$. The absolute value of the determinant can be interpreted geometrically as the area($2$-dimensional volume) spanned by the two column vectors $\begin{pmatrix} 1\\0 \end{pmatrix},\begin{pmatrix} 2\\1 \end{pmatrix} $. You can clearly see it in this graph:
We can generalize this idea further to $3$-dimensional vectors. The volume of the parallelepiped spanned by three vectors $v_1,v_2,v_3$ in $\mathbb{R}^3$ is the determinant of the matrix having the three vectors as columns.
As you probably know, a two dimensional vector in $\mathbb{R}^2$ is mathematically represented as a couple $(a_1,a_2)$: $a_1$ represents the projection on the first axis (usually referred to as $e_1$), while $a_2$ represents the projection on the second axis($e_2$). These two numbers uniquely identify a vector in the real two dimensional vector space. In $\mathbb{R}^3$, three coordinates are needed: these are the projections on the axes $e_1,e_2,e_3$(these are usually called $x,y,z$).
The answer is not really that amazing as you would expect: you just add another numbers to your vector. Those numbers represent the projections of our vector on the axes $e_4,e_5,\dots e_n$ which are unitary and mutually orthogonal, as the vectors $e_1,e_2,e_3$. The determinant of an $n \times n$ matrix can be now geometrically interpreted as the $n$-dimensional volume of a parallelepiped spanned by the $n$ column vectors of the matrix and shares many properties with "usual" volumes.