How to prove that rank of matrix $A$ equals rank of $A^T\overline{A}$ Let $A \in \mathbb{C}^{m\times n}$ be a $m\times n$-matrix over the complex numbers $\mathbb{C}$, for $n,m\ge 1$. 
How do you prove that the rank of matrix $A$ equals the rank of $A^T\overline{A}$?
Here $\overline{A}$ is denotes the conjugate matrix and $A^T$ the transpose. 
 A: Using Gauss elimination, one can find an invertible matrix $P$ such that 
$A^T=PJ$ is in echelon form. This means the matrix $J$ is in row echelon form, and it has $r$ non-trivial rows and $m-r$ trivial rows, where $r$ denotes the rank of $A$. So $J$ has the form 
$$J=\begin{pmatrix} C \\ 0 \end{pmatrix}$$
where $C$ is $r\times n$ matrix of rank $r$ in echelon form. 
Since the last $m-r$ rows of $J$ are $0$, the last $m-r$ columns of $\overline{J}^T$ are also zero. So $J\overline{J}^T$ has the form 
$$J\overline{J}^T=\begin{pmatrix} D & 0 \\ 0 & 0 \end{pmatrix} $$
where $D=C\overline{C}^T$ is $r\times r$ square matrix.  
Now, $A^T\overline{A}=PJ\overline{J}^T\overline{P}^T$. Since $r=\text{rank}(A)= \text{rank}(A^T)=\text{rank}(J)=\text{rank}(C)$, and $$\text{rank}(A^T\overline{A})=\text{rank}(J\overline{J}^T)=\text{rank}(C\overline{C}^T)=\text{rank}(D)$$ 
since $A^T\overline{A}$ it is obtained from $J\overline{J}^T$ by multiplying by invertible matrices, we are reduced to the case $A^T=J$ is in echelon form, even reduced to the case $A^T=C$ is in echelon form and with rank equal the number of rows. 
But $D$ is a $r\times r$ matrix, his rank is $\le r$, and to show the result we only need to show $\text{det}(D)\ne 0$, or, equivalently, that no (column) vector $v\in \mathbb{F}^{1,n}$, $v\ne 0$, verifies that $Dv=0$. But 
$$0=Dv=C\overline{C}^Tv=C(\overline{C}^Tv)$$
hence 
$$ 0 =\overline{v}C(\overline{C}^Tv)$$
If we denote by $w=\overline{v}C$, then $(\overline{C}^Tv)=\overline{w}^T$, which means 
$$0 =\overline{v}C(\overline{C}^Tv)=w\overline{w}^T=\langle w, w \rangle=| w|^2,$$
so $w=0$. This implies $$\overline{C}^Tv=\overline{w}^T=0$$
so $v=0$ since $\text{rank}(C)$ equals the number of rows $=r$. 
