Mathematics of transformation of 2-D to 1-D coordination.

Question

Let's see an example

In cartesian coordinate system:

x/y   0   1   2
    +---+---+---+
  0 | A | B | C |
    +---+---+---+
  1 | D | E | F |
    +---+---+---+
  2 | G | H | I |
    +---+---+---+

Then transform to this:

n   0   1   2   3   4   5   6   7   8
  +---+---+---+---+---+---+---+---+---+
  | A | B | C | D | E | F | G | H | I |
  +---+---+---+---+---+---+---+---+---+

where x, y, n are axes and A...I are just containing text on those positions

Is there any theory that can explain this kind of transformation in term of mathematics?

I just want to use it to explain on my work that something it doesn't change over this kind of transformation. And it much more convenient if I have some mathematical notations to explain about this.

Answer 1

You want transform square matrix into row vector.

Let

$M$ is your $matrix$ dim $3x3$
$i,j,k$ versors of standard basis dim $3x1$
$I$ identity matrix dim $3x3$,
$0$ - zero matrix dim $3x3$,

then

required steps are:

1.extract from matrix rows e.g. $row_1 ={i^T} M$
2.make space with dimension 9 using blocks from $0_{3x3}$ and $I_{3x3}$
3.place rows in proper positions of this space

all this is done by below formula which generates $1x9$ vector:

$v_{1x9}={i^T}_{1x3}M_{3x3} [I 0 0]_{3x9}+{j^T}_{1x3}M_{3x3} [0 I 0]_{3x9}+{k^T}_{1x3}M_{3x3} [0 0 I]_{3x9}$

.