Given a OLS regression problem $$ Y = \beta X + \epsilon $$ its analytical solution is given by $$ \hat \beta = (X^TX)^{-1}X^TY $$ What constraints should I put on X or how should I define/transform X so that, $$ (X^TX)^{-1}X^T \ge 0 $$ What will be the analytical solution of the resulting regression problem?
Note that X is non-square (M x N) matrix.