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How to generate random projection matrices?

Asked by Rufat Imanov on 24 Jul 2019
Latest activity Edited by Bruno Luong
on 26 Jul 2019
As it is said in the question, I am looking for a Matlab function that generates random projection matrices, so that I can use it for linear programming.

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3 Answers

Answer by KALYAN ACHARJYA on 24 Jul 2019
Edited by KALYAN ACHARJYA on 24 Jul 2019

function P=projection_mat(n)
A=colbasis(magic(n));
P=A*inv(A'*A)*A';
end
The colbasis function is here
Here n represent size of square matrix. Please note that I have answered this question from here
Command Window:
>> y=projection_mat(6)
y =
0.7500 -0.0000 0.2500 0.2500 -0.0000 -0.2500
-0.0000 1.0000 0.0000 -0.0000 -0.0000 0.0000
0.2500 0.0000 0.7500 -0.2500 -0.0000 0.2500
0.2500 -0.0000 -0.2500 0.7500 -0.0000 0.2500
-0.0000 -0.0000 -0.0000 -0.0000 1.0000 -0.0000
-0.2500 0.0000 0.2500 0.2500 -0.0000 0.7500
You can generate any size matries, just pass the same size matrix to colbasis function.
Hope it helps!

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I have a question, the function generates only the same matrices, so it does not generate random matrix, but generates a single solution
Is there any necessity having fixed size matrices?
>> y=projection_mat(6)
y =
0.7500 -0.0000 0.2500 0.2500 -0.0000 -0.2500
-0.0000 1.0000 0.0000 -0.0000 -0.0000 0.0000
0.2500 0.0000 0.7500 -0.2500 -0.0000 0.2500
0.2500 -0.0000 -0.2500 0.7500 -0.0000 0.2500
-0.0000 -0.0000 -0.0000 -0.0000 1.0000 -0.0000
-0.2500 0.0000 0.2500 0.2500 -0.0000 0.7500
>> y=projection_mat(5)
y =
1.0000 -0.0000 -0.0000 -0.0000 -0.0000
-0.0000 1.0000 -0.0000 -0.0000 -0.0000
-0.0000 -0.0000 1.0000 -0.0000 0.0000
-0.0000 -0.0000 -0.0000 1.0000 0.0000
-0.0000 -0.0000 -0.0000 -0.0000 1.0000
>>
yes, the size of matrix should be the same, but the matrix by itslef needs to change. Beacuse, I am working on finding a specific matrix that can project a square into square, so that the nodes/edges are not outside of the boundary of the square.

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Answer by Bruno Luong
on 25 Jul 2019
Edited by Bruno Luong
on 25 Jul 2019

n = 5
r = 3; % rank, dimension of the projection subspace
[Q,~] = qr(randn(n));
Q = Q(:,1:r);
P = Q*Q' % random projection matrix P^2 = P, rank P = r

  5 Comments

You are completely right, I have miststated my question, I need to generate an orthogonal projection matrix
Sorry I think the only projection matrix that is orthogonal is diagonal matrix with 1 or 0 on the diagonal. So there is no really randomness for what you ask.
I wonder if you mistaken "orthogonal projection matrix" and "projection matrix that is orthogonal". They are not the same.
Mine is "orthogonal projection matrix", which is projection matrix (P^2==P) that has additional properties
  1. symmetric
  2. all eigen values are 0 or 1.

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Answer by Image Analyst
on 25 Jul 2019

Not sure what you mean by projection, but the radon transform does projections. That's its claim to fame. It basically projects a matrix along any angle and gives you the sum of the interpolated values along the projection angle. This is the crucial function for reconstructing 3-D volumetric CT images from 2-D projections.
The radon() function requires the Image Processing Toolbox.

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