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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, ...).

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    $\begingroup$ Is en.wikipedia.org/wiki/Weyl%27s_inequality sufficient for you? $\endgroup$ – Sandeep Silwal Nov 8 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 at 18:58
  • $\begingroup$ Welcome to MathOverflow! I edited the wording of the question since the expression "weakly smaller than" for a non-strict inequality seems to be very uncommon. $\endgroup$ – Jochen Glueck Nov 8 at 19:04
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    $\begingroup$ Concerning the content of the question, do you assume that $c \ge 0$? $\endgroup$ – Jochen Glueck Nov 8 at 19:05
  • $\begingroup$ @JochenGlueck, thank you for your comment. I have eddited my orginal question, I hope it is now more clear. $\endgroup$ – Seb Nov 10 at 14:07

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