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

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!

[Walter Roberson]

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]

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

Answer 1

[Matt J]

Hints:

(1) Consider [null(A), null(B)]

(2) Consider [A;B]

Answer 2

[Bruno Luong]

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)