I want to know what could be the possible rank of a matrix, which is constructed from a same vector but have two repeating rows. Lets say I have a vector $$x=\begin{bmatrix}1 &a &1& b\end{bmatrix}^T$$ of complex quantities $a \&b \in C$and if I define a rank-1 hermitian matrix $X=xx^T$, there will be two dependent rows in the matrix (1st and third rows)
$$ \begin{bmatrix} 1 &a^*& 1& b^*\\ a &aa^*& a& ab^*\\ 1 &a^*& 1& b^*\\ b &ba^*& 1& bb^*\\ \end{bmatrix} $$
I am wondering if the presence of two dependent rows will effect the rank in some way or will i still get a rank-1 matrix after getting the solution (i.e., putting the values of a and b). Question might sound silly but my derivation is stuck on this point. Any help would be appreciated.
A rank one matrix is a matrix in which the dimension of the vector space, spanned by the rows, is one.
This means that for any two pairs of rows in a rank one matrix, one will be a multiple of the other.