1
$\begingroup$

We have an asymptotic analysis problem for the eigenvalue performance of the following random matrix: $H=\{h_{ij}\}_{N_r\times N_t}$,

where each entry $h_{ij}$ is with a probability $p$ to obey the Gaussian distribution $N(0,σ^2)$, and with a probability $1-p$ to be zero.

Then we have following questions

a)Can we use a specific distribution (pdf) to describe the entries of this $N_r \times N_t$ matrix as $h_{ij}\sim (1-p)\delta(0)+p\frac{1}{\sigma\sqrt{2\pi}}e^{-j\frac{x^2}{2\sigma^2}}\delta(x)$?

b)When $N_r,N_t\rightarrow \infty$, and $N_r/N_t\rightarrow \beta$ (determined value), does the eigenvalues’ PDF go to a deterministic distribution? If the answer is Yes, what is the explicit expression of this deterministic distribution?

For question b), one potentially useful tool maybe the Mar˘cenko-Pastur law in the random matrix theory, but I do not know if it can be used indeed.

$\endgroup$
  • $\begingroup$ The matrices aren't square. Do you mean singular values instead of eigenvalues? $\endgroup$ – Lior Silberman Sep 29 '14 at 13:06
  • $\begingroup$ Yes, for a general case, singular values should be considered, and the Marchenko-Pastur law may can be used. But from my comparison between the asymptotic results and the numerical results, there is a mismatch, I cannot explain the reason. $\endgroup$ – Wenlong Cai Sep 30 '14 at 0:12
2
$\begingroup$

I don't really understand your question (a) - but if you look up "mixtures", it will be answered. As for question (b), the magic words are "Girko's circular law", and the magic reference is the Wikipedia.

$\endgroup$
  • $\begingroup$ For question(a), I think the entries follow the distribution something like the Gaussian-Bernoulli distribution in the compress sensing applications. For question (b), I do not quite sure if the Mar˘cenko-Pastur law in the random matrix theory can be used indeed. $\endgroup$ – Wenlong Cai Sep 29 '14 at 3:19
  • 1
    $\begingroup$ Marchenko-Pastur is about SINGULAR values, and has nothing to do with this question. I have answered your question. $\endgroup$ – Igor Rivin Sep 29 '14 at 3:40
  • $\begingroup$ Thanks for your help, I will check your reference. I have another question about the circular law, does the circular law suit for all the distributions, or there is a limitation on the distribution of the entries? $\endgroup$ – Wenlong Cai Sep 29 '14 at 4:11
  • $\begingroup$ @WenlongCai that is discussed in the wikipedia article, with references (in particular, to the Tao-Vu paper) $\endgroup$ – Igor Rivin Sep 29 '14 at 4:16
  • $\begingroup$ I run a simulation on this kind of sparse random matrix, and compared the eigenvalues' PDF simulation results with the asymptotic results from the random matrix book, there is mismatch between these results. I will double check the simulation program. $\endgroup$ – Wenlong Cai Sep 29 '14 at 6:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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