Hi,


i have two questions that seem to bother me lately. Maybe you could help me and/or point me in any related literature.


1) Assume hermitian matrix $H \in \mathcal{C}^{n \times n}$ that has rank $r$. How many real valued parameters are needed to describe it?


2) Now assume that you also have the knowledge that $H = G^\*G$ for some $G \in \mathcal{C}^{n \times m}$ with rank r. Does this piece of information reduce the number of real valued parameters needed to specify H?


Thank you very much for your help,


Alex



Number of parameters needed to specify a Hermitian matrix of rank r.
The answer to 1) is simple. An Hermitean matrix is determined by its eigenvalues and
eigenspaces. Assign $r$ distinct eigenvalues ($r$ real parameters). Their
eigenspaces are orthogonal lines. One line depends on $n-1$ complex parameters, the next line
must be perpendicular to the first, so it depends on $n-2$ complex parameters, and so on. 
Summing up, we obtain
$$r+2((n-1)+...+(n-r))=2nr-r^2$$
real parameters. To check, set $r=n$, we get $n^2$, the real dimension of the space of all
Hermitean matrices. Set $n=2,r=1$ we get $3$.


The answer to 2) is even simpler, because every Hermitean matrix can be represented in this
form. 


