Kernel of Convex Combination of Matrices

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

[ αA + (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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1 thought on “Kernel of Convex Combination of Matrices

  1. 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

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