$B^HB=b$ number of solutions

Question

Let B any non square real matrix $m\times n$ with linearly independent columns.

$B^HB=b$ (where $B^H$: conjugate transpose of Β) has only one solution for every $b\in R^m\neq0 $ (1)

Is (1) true or false and why?

Answer 1

Notice that $B$ is injective since it has a linearly independent columns and then $$\langle B^HB x,x\rangle=\langle B x,Bx\rangle=||Bx||^2=0\iff Bx=0\iff x=0$$ so $B^HB$ is symmetric definite positive matrix hence it's invertible. Conclude.