2*2 Matrix factorization/decomposition
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Hi all, this is my first forum post on this community.
For my high-school essay relating to mathematics, I am looking at a specific problem relating to affine transformations and geometry. Using the affine2d and imwarp method, I was able to transform a coloured image according to a customised 3 by 3 matrix. The matrix I used was:
1 3 0
3 -4 0
1 1 1
I want to find out the combinations of shearing, scaling and rotation that eventually led to my linear mapping. Specifically, I want to find out what combination of the matrices listed below:
[1,0;
0,1],
[1,1;
0,1],
[1,0;
1,1],
and:
[0,1;
1,0],
Resulted in my linear mapping matrix of:
1 3
3 -4
which was used to transform my image.
I did some research and found out it should be possible, as all invertible matrices can be factorised to elementary matrices.I also found out what I wanted may have something to do with PLU(permuted lower upper decomposition), but am clueless to how to implement it.
Apologies if anything I wrote is incorrect or inaccurate. Any help would be greatly appreciated.
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Matt J
2021년 8월 1일
편집: Matt J
2021년 8월 1일
Specifically, I want to find out what combination of the matrices listed below:
It is not clear in what sense the matrices are to be "combined". If you mean simply to multiply them together, note that because each of these matrices has determinant = +/-1, combinations of them multiplied together will also have determinant = +/-1. This means that a matrix like
1 3
3 -4
whos determinant is -13 cannot be decomposed this way.
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