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For any positive matrix $A$, let $M=\frac{A}{\rho(A)}$. $G$ is the Perron projection of $M$.

I happen to find for any positive vector $v>0$, there is always $\left| \Vert v\Vert_1-\Vert vG\Vert_1 \right|\geq \left| \Vert vM\Vert_1-\Vert vG \Vert_1 \right|.$ But have no ideas why...

Thanks for any advice!

WASABI
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  • What do you mean by "I happen to find"? – Jochen Glueck Dec 17 '20 at 22:50
  • Alright, I just checked the inequality by hand for a certain class of very "non-symmetric" $2 \times 2$-matrices which I would have considered as "canonical" candidates for a counterexample. But the inequality turns out to be correct for them. I would really love to know how you came up with this inequality, and why you suspect that it holds. – Jochen Glueck Dec 17 '20 at 23:22
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    @JochenGlueck Thank you very much for great thoughts! Many many apologies, as my project was ceased, I have forgotten this question for months. This question was found in my research on discrete fractals. And interestingly, according to certain patterns of fractals, plenty of similar questions (or say, conjecture) can be proposed. But it is really frustrating that I fail to find relevant references working on such stuff. () – WASABI Jul 21 '21 at 13:04

1 Answers1

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This is not true. Here is a random counterexample. Let \begin{aligned} &A=\pmatrix{2&1&2\\ 2&2&1\\ 1&2&1}, \ v=\pmatrix{2\\ 1\\ 1},\\ &M=\pmatrix{ 0.42457&0.21229&0.42457\\ 0.42457&0.42457&0.21229\\ 0.21229&0.42457&0.21229}, \ G=\pmatrix{ 0.37543&0.35924&0.29917\\ 0.39235&0.37543&0.31265\\ 0.31265&0.29917&0.24914}. \end{aligned} Then \begin{aligned} &|\|v^T\|_1-\|v^TG\|_1| - |\|v^TM\|_1-\|v^TG\|_1|\\ =\,&|4-4.00906|-|4.03344-4.00906|\\ =\,&0.00906-0.02438\\ <\,&0. \end{aligned} However, I don't find any counterexamples when $A$ is $2\times2$.

user1551
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    +1. (Since science is in need of more replication, I've redone all the computations in your answer (using Octave), and got the same results.) – Jochen Glueck Dec 18 '20 at 12:47
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    Really appreciate your and Jochen's computations!!! Many many apologies, as my project was ceased, I have forgotten this question for months. It seems very likely that for 2 x 2 it is true, which makes this question trivial though. By the way, this question was found in research on discrete fractals, and according to some fractals, many similar problems can be raised up, just for your interests :) – WASABI Jul 21 '21 at 13:09