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!