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!


One 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


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