2*2 Matrix factorization/decomposition

Question

Aaron Zheng 2021-7-22

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编辑： Matt J 2021-8-1

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 2021-8-1

编辑：Matt J 2021-8-1

在 MATLAB Online 中打开

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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Answer 1

Matt J 2021-7-22

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https://ww2.mathworks.cn/matlabcentral/answers/883704-2-2-matrix-factorization-decomposition#answer_751679

编辑：Matt J 2021-7-22

在 MATLAB Online 中打开

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.

Easy enough to do with the lu() command:

 [L,U,P]=lu([1 3;3 -4])
L = 2×2
    1.0000         0
    0.3333    1.0000
U = 2×2
    3.0000   -4.0000
         0    4.3333
P = 2×2
     0     1
     1     0

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Aaron Zheng 2021-8-1

But the U matrix [3.0 -4; 0 4.3333] is not a perfect shear. The diagonal values(3 and 4.333) are not equal to each other. Also L*U does not equal A, instead it equals P*A, which is a different matrix from A. Is it possible to get L*U to equal A instead?

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