For linear equalities A x = b, the solution can be written as x = ( A T A ) − 1 A T b.For linear inequalities system A x ≥ b, on what condition the solution is x ≥ ( A T A ) − 1 A T b?

Question

For linear equalities   A  x  =  b, the solution can be written as   x  =  (      A    T    A      )          −      1            A    T    b.For linear inequalities system   A  x  ≥  b, on what condition the solution is   x  ≥  (      A    T    A      )          −      1            A    T    b?

mseralge · Accepted Answer

I think you have a slight misconception. The solution of   A  x  =  b is   x  =      A          −      1        b, assuming   A is invertible. If   A is not invertible,   x  =  (      A    T    A      )          −      1            A    T    b gives a least-squares solution that does not necessarily satisfy   A  x  =  b. As for the inequality case, the solution set is identical to   x  ⪰      A          −      1        b or   x  ⪰  (      A    T    A      )          −      1            A    T    b when   A is diagonal and positive definite, but probably not otherwise.

For linear equalities A x = b , the solution can be written as x = (

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