Consider U = [ u 1 . . . u N ] to be a...

Augustus Acevedo

Augustus Acevedo

Answered

2022-07-15

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 , i
Is there a non-graphical, non-numerical approach that I can take to obtain all possible a?

Answer & Explanation

Jamarcus Shields

Jamarcus Shields

Expert

2022-07-16Added 17 answers

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 } .

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