Way to solve AX=XB

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SATISH SONWANE
SATISH SONWANE on 25 Jan 2023
Edited: Bruno Luong on 28 Jan 2023
Is there any implementation of Tsai and lenz's (Or any other) method for solving AX=XB for hand- Eye Calibration?

Answers (3)

the cyclist
the cyclist on 25 Jan 2023
This is a special case of the Sylvester equation.
Looks like the sylvester function will be helpful for you.
You might also be interested in this blog post on the topic by Cleve Moler.

Torsten
Torsten on 28 Jan 2023
Edited: Torsten on 28 Jan 2023
dim = 4;
X = sym('X',[dim dim]);
A = rand(dim);
B = A.';
[M, ~] = equationsToMatrix(A*X==X*B)
if rank(M) < size(A,1)^2
N = null(M);
for i = 1:size(N,2)
S{i} = reshape(N(:,i),size(X));
S{i}
A*S{i}-S{i}*B
end
end

Matt J
Matt J on 28 Jan 2023
Edited: Matt J on 28 Jan 2023
[ma,na]=size(A);
[mb,nb]=size(B);
%size(X)=[na,mb]
X=null( kron(speye(mb),A) - kron(B.',speye(na)) );
X=reshape(X,na,mb,[]);
  2 Comments
the cyclist
the cyclist on 28 Jan 2023
I couldn't get this method to work. Am I overlooking something dumb?
rng default
A = rand(5);
B = rand(5);
[ma,na]=size(A);
[mb,nb]=size(B);
X=null( kron(speye(mb),A) - kron(B.',speye(na)) );
Error using svd
SVD does not support sparse matrices. Use SVDS to compute a subset of the singular values and vectors of a sparse matrix.

Error in null (line 75)
[V, s] = svd(A','vector');
X=reshape(X,na,mb,[]);
Bruno Luong
Bruno Luong on 28 Jan 2023
Edited: Bruno Luong on 28 Jan 2023
null can only work wth full matrix
rng default
A = rand(5);
XX = rand(5);
B = XX\(A*XX);
[ma,na]=size(A);
[mb,nb]=size(B);
K=null( kron(eye(mb),A) - kron(B.',eye(na)));
R = rand(size(K,2),1); % Any random vector with this size will do the job
X = reshape(K*R,[na,mb])
X = 5×5
0.1100 0.1626 0.0772 -0.0458 0.0810 -0.0536 -0.3884 0.0510 0.0552 -0.0602 0.1937 -0.1993 0.3240 0.3508 -0.0352 -0.1165 0.4388 -0.3413 -0.3089 0.1851 0.0500 0.0277 0.0913 0.1389 0.0450
norm(A*X-X*B)
ans = 7.4501e-16

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