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I know I can bound the spectral radius of a matrix by a norm of the matrix. Are there other approaches? In my case, I have a positive square matrix J(v) that depends on some fixed parameters and the vector v that lives in the simplex (i.e., $v_i\geq0,\forall i$ and $\sum_i v_i =1$), and I need to show that $$\max_v \rho(J(v)) \leq 1.$$

Simulations reveal that this is true, but also that well known norms (i.e., infinity norm or spectral norm) don't give me the right bounds. Is proof by contradiction useful? I am going to leave this question vague because the details are somewhat complicated and probably not of much interest here, but suggestions and examples would be most welcomed.

Andres
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  • How explicit are your matrices on which your upper bounds are not tight? It might help to reduce the set of maximizers $v$ to consider using, say, convexity properties of your parametrization – md5 Jun 29 '21 at 17:00
  • @A When you say that $J(v)$ is a "positive" matrix, does that mean that all of its entries are positive or is positive synonymous with "positive definite"? – Ben Grossmann Jun 29 '21 at 17:03
  • @A If you mean the former, then the following might be useful: for any positive matrix $A$, there exists a diagonal matrix $D$ with positive diagonal entries such that $\rho(A) = |DAD^{-1}|_\infty$ – Ben Grossmann Jun 29 '21 at 17:05
  • @A We have no way to answer a question like "is proof by contradiction useful?" without knowing more about the problem – Ben Grossmann Jun 29 '21 at 17:07
  • @BenGrossmann $J(v)$ has all positive entries, thanks for that suggestion. – Andres Jun 29 '21 at 17:29
  • @BenGrossmann I have created a question with all the details of my particular problem here: https://math.stackexchange.com/q/4186355/165163 I fear it may be too specific to generate interest here -- perhaps there is a better way of asking this quuestion? – Andres Jun 29 '21 at 19:02
  • @md5 I have created a question with all the details of my particular problem here: math.stackexchange.com/q/4186355/165163 I fear it may be too specific to generate interest here -- perhaps there is a better way of asking this quuestion? – Andres Jun 29 '21 at 19:03

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