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?
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2 
- $\begingroup$ What is the size of $B^H B$? What is its rank? $\endgroup$user114628– user1146282014-02-15 17:46:27 +00:00Commented Feb 15, 2014 at 17:46
- $\begingroup$ No info has been given for $B^HB$... $\endgroup$DimitrisD– DimitrisD2014-02-15 18:06:02 +00:00Commented Feb 15, 2014 at 18:06
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1 Answer
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0 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.