Number of parameters needed to specify a Hermitian matrix of rank r. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T14:26:04Z http://mathoverflow.net/feeds/question/110189 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110189/number-of-parameters-needed-to-specify-a-hermitian-matrix-of-rank-r Number of parameters needed to specify a Hermitian matrix of rank r. Kostas 2012-10-20T20:35:29Z 2012-10-20T21:39:43Z <p>Hi,</p> <p>i have two questions that seem to bother me lately. Maybe you could help me and/or point me in any related literature.</p> <p>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?</p> <p>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?</p> <p>Thank you very much for your help,</p> <p>Alex</p> http://mathoverflow.net/questions/110189/number-of-parameters-needed-to-specify-a-hermitian-matrix-of-rank-r/110194#110194 Answer by Federico Poloni for Number of parameters needed to specify a Hermitian matrix of rank r. Federico Poloni 2012-10-20T21:17:22Z 2012-10-20T21:17:22Z <p>Not sure if I am missing something here...</p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/110189/number-of-parameters-needed-to-specify-a-hermitian-matrix-of-rank-r/110195#110195 Answer by Alexandre Eremenko for Number of parameters needed to specify a Hermitian matrix of rank r. Alexandre Eremenko 2012-10-20T21:18:37Z 2012-10-20T21:39:43Z <p>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 $3$.</p> <p>The answer to 2) is even simpler, because every Hermitean matrix can be represented in this form. </p> http://mathoverflow.net/questions/110189/number-of-parameters-needed-to-specify-a-hermitian-matrix-of-rank-r/110198#110198 Answer by Felix Goldberg for Number of parameters needed to specify a Hermitian matrix of rank r. Felix Goldberg 2012-10-20T21:29:44Z 2012-10-20T21:29:44Z <p>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.</p>