# Transformation matrix for matrix indices to cartesian coordinates

In MatLab matrices, the indices are as follows:

(1,1) (1,2) (1,3)
(2,1) (2,2) (2,3)
(3,1) (3,2) (3,3)


This is an example 3x3 matrix. In corresponding cartesian coordinate system, the representation would be:

(-1,1) (0,1) (1,1)
(-1,0) (0,0) (1,0)
(-1,1) (0,-1) (1,-1)


Say, I have any square matrix with dimension-N, where N is odd. I need a generic transformation matrix such that I can get a vector as cartesian coordinates from matrix indices. Does such a function already exist? How should I go ahead in implementing this?

Thanks.

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I think your $(-1,1)$ on the second matrix, first column, third row should be $(-1,-1)$. – a.r. Jan 19 '12 at 4:21

$$(x,y) = f(i,j) = \left( j-\frac{n+1}{2} ,-i + \frac{n+1}{2}\right) \ .$$
Here $i$ is the index for the rows, $j$ the one for the columns and $n$ the order of your square matrix.
Interchange indices $i$ and $j$ in initial matrix, then flip it upside down to get the same orientation like a usual coordinate system and then subtract $(2,2)$ or $(\frac{n+1}{2},\frac{n+1}{2})$ in general to shift the center.