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Let $M$ be a matrix such that $\forall i,j$ $M_{ij}\geq 0$ and suppose that $M$ is irreductible.

1 - Is there a natural change of basis such that the new matrix became strictly positive : $\forall i,j$ $M_{ij}’>0$?

2 - Can this technique be generalised to non negative operator $P$ (in infinite dimension) ? For example, take $P(f)=1_{[-1/2,1-2]}\star f$

Motivation : I would like to use the theorem of Birkhoff Hopf that state that the matrix (operator) is strictly contracting for the Hilbert metric on the projective space. But it cannot be used if $\exists i,j,M_{ij}=0$.

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    $\begingroup$ A certain power of your operator will contract the Hilbert metric. Isn't this enough? $\endgroup$
    – Jairo Bochi
    Commented Sep 21, 2017 at 13:27
  • $\begingroup$ natural change of basis : $M'=GMG^{-1}$. $\endgroup$
    – RaphaelB4
    Commented Sep 21, 2017 at 13:36
  • $\begingroup$ @JairoBochi , no it is not enough. I don't tell you about the second step but it is with different matrices $M,N,..$ and I would like to have the contracting property for a product of these matrices. $\endgroup$
    – RaphaelB4
    Commented Sep 21, 2017 at 13:43
  • $\begingroup$ Notice that an obvious obstruction to a strictly positive conjugate is trace$(M) = 0$. In addition, for the infinite case, strict positivity is extremely problematic. $\endgroup$
    – David Handelman
    Commented Sep 21, 2017 at 13:44
  • $\begingroup$ @JairoBochi well, you need primitivity for that however you can add the identity to get such property from irreducibility. $\endgroup$
    – Surb
    Commented Sep 21, 2017 at 14:04

1 Answer 1

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A positive matrix is primitive and so it has only one nonzero eigenvalue of maximum modulus.

An irreducible which is not primitive has (by definition) strictly more than one eigenvalue with maximum modulus.

As basis change leave eigenvalues invariant, this shows that such change of basis might not exists if we only assume irreducibility.

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  • $\begingroup$ Corresponding results can be found in Horn & Johnson's book, chapter 8. $\endgroup$
    – Surb
    Commented Sep 21, 2017 at 14:17

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