1
$\begingroup$

Let $W = S + cT$, where $|c| \le 1$ is a real constant and where $S$ and $T$ are square matrices containing real numbers from the interval $[0,1]$.

Assume moreover that the all eigenvalues $\lambda$ of $S$ and $T$ satisfy $|\lambda| \le 1$. (Both matrices are right stochastic in that each row sums to one.)

Is it possible to derive an informative bound for the spectral radius of $W$? Besides being right stochastic, the matrices $S$ and $T$ do not have a special structure (e.g. Hermitian, diagonalizable, ...).

Improve this question
$\endgroup$
9
  • 1
    $\begingroup$ Is en.wikipedia.org/wiki/Weyl%27s_inequality sufficient for you? $\endgroup$
    – Sandeep Silwal
    Nov 8, 2019 at 18:51
  • $\begingroup$ @SandeepSilwal: According to the OP, the matrices $S$ and $T$ are not assumed to be Hermitian. $\endgroup$
    – Jochen Glueck
    Nov 8, 2019 at 18:58
  • 2
    $\begingroup$ Concerning the content of the question, do you assume that $c \ge 0$? $\endgroup$
    – Jochen Glueck
    Nov 8, 2019 at 19:05
  • 1
    $\begingroup$ Could the answer perhaps be that $A=(1+c)^{−1}W$ is again right stochastic, such that all eigenvalues $\mu$ of $A$ also satisfy $|\mu|≤1$ and all eigenvalues $\lambda$ of $W$ are therefore $|\lambda|≤(c+1)$? $\endgroup$
    – Seb
    Nov 10, 2019 at 14:25
  • 1
    $\begingroup$ @Seb: If $c \in [0,1]$, then $(1+c)^{-1}W$ is again right stochastic. But if $c \in (-1,0]$, then $W$ can, of course, have negative entires. $\endgroup$
    – Jochen Glueck
    Nov 11, 2019 at 7:21

1 Answer 1

Reset to default
-1
$\begingroup$

On the zero-level, treat both $S$ and $T$ as random Ginibre matrices (not even real, just members of the GinU ensemble). Also, neglect that the spectral radius of both is one & at least one eigenvalue is exactly 1. Then, if both you right stochastic matrices are sampled randomly, their spectral bulks are discs of the radius $1/\sqrt{N}$

Then the spectral bulk of $W$ is the disc of the radius $\sqrt{\frac{1+c^2}{N}}$.

Improve this answer
$\endgroup$
6
  • $\begingroup$ I don't think he question is asking for expected bounds on random matrices. The term "stochastic matrix" appears to be this one en.wikipedia.org/wiki/Stochastic_matrix and so the Ginibre ensemble is not a good model. (In particular, note that the matrix entries of S and T are required to lie in [0,1].) $\endgroup$
    – Yemon Choi
    Dec 16, 2023 at 23:32
  • $\begingroup$ @Yemon Choi From the perspective of the support of the spectral bulk (which basically includes all eigenvalues expect the leading one, $\lambda_1 =1$), a random stochastic matrix $S$ is a real Ginibre matrix with variance of its elements equal $\sqrt(N)$. The support is not sensitive to the features of $S$ such as normalization etc. $\endgroup$
    – trurl
    Dec 18, 2023 at 17:54
  • $\begingroup$ Regardless of the claimed universality features, the original question seems to be asking for worst-case bounds rather than that what is true "generically". $\endgroup$
    – Yemon Choi
    Dec 18, 2023 at 21:24
  • $\begingroup$ I am also not entirely convinced about the universality claims. Stochastic matrices never have negative entries, and they are constrained to have constant row sums without any corresponding constraint on the columns. I don't see why Gaussian random matrices would provide a good model. (Your claim about support of the spectral bulk certainly must depend on the mean value of each random entry, otherwise I could simply add a constant to everything.) $\endgroup$
    – Yemon Choi
    Dec 18, 2023 at 21:30
  • $\begingroup$ See this arxiv.org/abs/0812.0567 (for random Markov chains) and this arxiv.org/abs/0804.2361 (for random quantum operations) $\endgroup$
    – trurl
    Dec 19, 2023 at 10:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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