Consider U = [ u 1 . . . u N ] to be a non-singular square matrix. I am interested in range space of U but with one caveat, I want to find all a such that each element of U a is non-negative. In other words for all natural basis vectors e i 0 ≤ e i T U a , ∀ iIs there a non-graphical, non-numerical approach that I can take to obtain all possible a?

Question

Consider   U  =  [      u    1    .  .  .      u    N    ] to be a non-singular square matrix. I am interested in range space of   U but with one caveat, I want to find all   a such that each element of   U  a is non-negative. In other words for all natural basis vectors       e    i    0  ≤      e    i    T    U  a  ,  ∀  iIs there a non-graphical, non-numerical approach that I can take to obtain all possible   a?

Jamarcus Shields · Accepted Answer

Let         R    +    N    =  {  x  =  (      x    1    ,  …  ,      x    N    )  ∈        R    N    :      x    i    ≥  0       for each     i  }  =  {      λ    1        e    1    +  ⋯  +      λ    N    e  N  :      λ    i    ≥  0       for each     i  }  .Since the matrix   U is non-singular,      U          −      1        (        R    +    N    )  =  {      λ    1        U          −      1            e    1    +  ⋯  +      λ    N        U          −      1            e    N    :      λ    i    ≥  0       for each     i  }  .

Consider U = [ u 1 </msub> . . . u N </msub>

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