3

Could you suggest me a book where I can find a proof of Perron-Frobenius theorem (especially for nonnegative matrices) based on a Brouwer fixed point theorem?

Gabriela Bałazy
  • 31
  • 2
    You shouldn't need to refer a book. Try applying Brouwer's fixed point theorem on the portion of the unit ball in the positive quadrant/octant (i.e. all coordinates $> 0$) and consider the map $v \mapsto Av/||Av||$. What does a fixed point here imply? – Osama Ghani Jun 07 '21 at 22:08
  • The problem is that I need this book to my graduate work. – Gabriela Bałazy Jun 07 '21 at 22:14
  • @OsamaGhani Don't we need a compact set for Brouwer? – saulspatz Jun 07 '21 at 22:19
  • I don't want to prove by this way. I have already proved it by using graphs. Now I'm looking for some informations about different ways and different proofs and note where I can find it. – Gabriela Bałazy Jun 07 '21 at 22:28
  • 1
    @saulspatz sorry I meant all coordinates $\geq 0$. – Osama Ghani Jun 07 '21 at 23:42

1 Answers1

1

Since the OP is, according to one of her comments, looking for different proofs of the Perron-Frobenius theorem, it might be wortwhile to point out the following survey paper:

C. R. MacCluer: The many proofs and applications of Perron’s theorem, SIAM Rev. 42, No. 3, 487-498, 2000 (link to zbMATH).

While the paper deals mainly with matrices with positive entries, it contains a large list of references, and surely many of them also deal with non-negative matrices. For instance, a long list of references to proofs by means of Brouwer's fixed point theorem can be found on page 494.

Jochen Glueck
  • 909