Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I had two doubts

  • Let $A$ be a hermitian matrix, what is the set of all unit norm vectors such that $x^{H}Ax\geq 0$.
  • Let $A_1$,$A_2$ be 2 hermitian matrices, what is the set of all unit norm vectors such that $x^{H}A_1 x\geq 0$ and $x^{H} A_2 x\geq 0$.
share|improve this question
1  
For the first question, if you diagonalize the matrix you can at least write up a simple equation on the form $\sum\lambda_j\lvert x_j\rvert^2\ge0$, where the $\lambda_j$ are the eigenvalues. Keep them in order to get a better handle on what that means. The second question is harder, unless the two matrices commute of course. –  Harald Hanche-Olsen Nov 7 '12 at 16:58

1 Answer 1

up vote 1 down vote accepted

I am not sure what exactly do you expect, but basically every answer to your question is just a reformulation of the required point sets. Suppose we are talking about vectors in $\mathbb{C}^n$. Let $U$ be a unitary matrix such that $UAU^H = \mathrm{diag}(d)$ for some $d\in\mathbb{R}^n$ ($d$ is a real vector because every eigenvalue of a Hermitian matrix is real). Define a polytope $P$ and a halfspace $H$ by \begin{align} P &= \left\{x=(x_1,\ldots,x_n)^T\in\mathbb{R}^n: x_i\ge 0\ \forall i,\ \sum_kx_k= 1\right\},\\ H &= \left\{x\in\mathbb{R}^n: x^Td\ge 0\right\}. \end{align} Note that $u=(u_1,\ldots,u_n)\in\mathbb{C}^n$ is a unit vector if and only if $$ f(u)\equiv u\circ \overline{u}\equiv (|u_1|^2,\ldots,|u_n|^2)\in P. $$ Then $$ \left\{x\in\mathbb{C}^n: \|x\|=1,\ x^HAx\ge 0\right\} = U^H\left(f^{-1}(H\cap P)\right). $$ This is because \begin{align} x^HAx\ge 0 &\Leftrightarrow (Ux)^H\mathrm{diag}(d)(Ux) \ge 0\\ &\Leftrightarrow y^Td \ge 0,\ \textrm{ where } y=f(Ux)\\ &\Leftrightarrow y\in H\cap P\\ &\Leftrightarrow Ux\in f^{-1}(H\cap P)\\ &\Leftrightarrow x\in U^H\left(f^{-1}(H\cap P)\right). \end{align} So, the set $\{x\in\mathbb{C}^n: \|x\|=1,\ x^HA_1x, x^HA_2x\ge0\}$ would be something like $U_1^H\left(f^{-1}(H_1\cap P)\right) \cap U_2^H\left(f^{-1}(H_2\cap P)\right)$, but I wonder if this formulation could be useful.

share|improve this answer
    
its a really good answer from a optimization point, you essentially reformulated it as intersections of convex sets I believe., but I was expecting something in the line of subspaces. But now I realize it will not be a subspace. –  dineshdileep Nov 8 '12 at 3:23
    
@dineshdileep: While $H\cap P$ is convex, $f^{-1}(H\cap P)$ is not. So I guess this formulation doesn't help :-( –  user1551 Nov 8 '12 at 3:58
    
your formulation is indeed useful!! –  dineshdileep Nov 21 '12 at 17:50

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.