## Number of parameters needed to specify a Hermitian matrix of rank r.

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

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 Look up "determinantal variety"... – Francois Ziegler Oct 20 at 21:18

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.

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 Regarding (2) - you must have meant every Hermitian positive definite matrix. – Felix Goldberg Oct 20 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 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.

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 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 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.

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 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 at 21:36 @Federico: Very possible, but perhaps the OP will weigh in... – Felix Goldberg Oct 20 at 22:00