# Spectral radius sum of two matrices

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, ...).

• Is en.wikipedia.org/wiki/Weyl%27s_inequality sufficient for you? – Sandeep Silwal Nov 8 at 18:51
• @SandeepSilwal: According to the OP, the matrices $S$ and $T$ are not assumed to be Hermitian. – Jochen Glueck Nov 8 at 18:58
• 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. – Jochen Glueck Nov 8 at 19:04
• Concerning the content of the question, do you assume that $c \ge 0$? – Jochen Glueck Nov 8 at 19:05
• @JochenGlueck, thank you for your comment. I have eddited my orginal question, I hope it is now more clear. – Seb Nov 10 at 14:07