Let $A$ ( $\mathbb{C}^{n \times n}$ matrix ) be lower triangular and satisfy $AA^* = I - bb^*$ where $b$ is a $\mathbb{C}^n$ matrix
Such $A$ is called a (lower) triangular input balanced. Show that for a given $b$, there is always a solution $A$ of this equation with non-negative diagonal elements ($A_{kk} \geq 0$ ).
Give an expression for the elements of this solution $A$ in terms of the elements of $b$.
Also I noticed , taking $b$ = [1 2]T , there are no solutions for the equation
Can someone please help with the solution?