Let **A** and **B** be two real *m*×*n* matrices, with *m* << *n*, such that there exists vectors **x**, **y** of length *n* such that

**Ax** = 0, **x** > 0

**By** = 0, **y** > 0

I.e., such that the kernel of each matrix contains a vector with all components larger than zero.

Question: Is the same true for convex combinations of **A** and **B**?

Put differently: For any 0 < α < 1, is there a vector **z** such that

[ α**A** + (1−α)**B** ] **z** = 0, **z** > 0 ?

I would be very thankful for a proof, or a refutation.

Thanks to anyone who can help!

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maybe this is too simplistic and not entirely what you are after, but if

(cA + (1-c)B)z = 0

one could argue that z has to lie in the nullspace of A and in the nullspace of B.

The question then is, if both nullspaces intersect. Aparently, there are different

methods for checking / computing the intersection of two nullspaces …

all the best

Christian