Check if nullspace is contained in another + finding intersection of 2 nullspaces.

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1) How to check (return true) if the nullspace of A is contained in the nullspace of B?
2) How to find the intersection of nullspaces of A and B?
Thanks!
  2 Comments
Walter Roberson
Walter Roberson on 27 Feb 2021
normalize each null vector. ismembertol 'byrows' of the transpose of the null spaces. (Convention is that the null vectors are presented as columns.)
Matt J
Matt J on 27 Feb 2021
Edited: Matt J on 27 Feb 2021
Unfortunately, ismembertol will not work. As the following example shows, the basis vectors returned by null(A) need not be a subset of the basis vectors returned by null(B), even if the nullspace of A is contained in the nullspace of B.
A=[-1 -1 1; 1 1 1];
B=[0,0,1];
null(A).'
ans = 1×3
0.7071 -0.7071 -0.0000
null(B).'
ans = 2×3
0 1 0 -1 0 0

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Answers (2)

Matt J
Matt J on 27 Feb 2021
Edited: Matt J on 27 Feb 2021
Hints:
(1) Consider [null(A), null(B)]
(2) Consider [A;B]

Bruno Luong
Bruno Luong on 27 Feb 2021
Edited: Bruno Luong on 27 Feb 2021
First question:
KA = null(A);
KB = null(B);
% Check span KA is included in span KB
PKA = KB*(KB'*KA); % Projection KA on span KB
ResA = PKA-KA; % Projection KA on orthogonal span KB
tol = 1e-9*sqrt(size(KA,1));
KAinKB = all(vecnorm(ResA,2,1)<tol)
The itersection has basis
null([A; B])
So you can also find the firs question by
size(null([A; B]),2) == size(null(B),2)

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