Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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

share|improve this question
    
Look up "determinantal variety"... –  Francois Ziegler Oct 20 '12 at 21:18

3 Answers 3

up vote 4 down vote accepted

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.

share|improve this answer
    
Regarding (2) - you must have meant every Hermitian positive definite matrix. –  Felix Goldberg Oct 20 '12 at 21:25
    
I had initially posted a follow-up question, but I solved it, so it wasnt needed it to keep it here. Thank you for your answer! –  Kostas Oct 21 '12 at 0:24

Not sure if I am missing something here...

1) Rank-$r$ Hermitian matrices are determined uniquely by their image $U$ and how they act when restricted to $U$. The image can be any dimension-$r$ subspace. Almost all subspaces have a basis in the form $\begin{bmatrix}I_r \\\\ X \end{bmatrix}$, with $X$ any $(n-r)\times r$ matrix, so you have $2(n-r)r$ degrees of freedom for the image. Possible actions on $U$ are isomorphic to $r\times r$ Hermitian matrices, so $r(r-1)+r=r^2$ real dof's. Overall this makes $2nr-r^2$ parameters.

2) If you know that $H$ is spd, then you have to restrict the second part to positive-definite matrices, but they still have the same number of parameters, so you still get the same answer.

share|improve this answer
    
Thank you for your quick reply. The answer is the same as the one below but it uses a different argument. Thanks –  Kostas Oct 21 '12 at 0:23

The question seems a bit vague - what is a "parameter"? In one sense the answer is $r(n+1)$, since if $H$ is Hermitian, then $H=\sum_{i=1}^{r}{\lambda_{i}x_{i}x_{i}^{T}}$. Each $x_{i}$ has $n$ entries.

share|improve this answer
    
I take it as "the dimension of the variety of rank-$r$ Hermitian matrices seen as embedded in $\mathbb{R}^{2n^2}$". –  Federico Poloni Oct 20 '12 at 21:36
    
@Federico: Very possible, but perhaps the OP will weigh in... –  Felix Goldberg Oct 20 '12 at 22:00

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.