Does every polynomial with a Perron root have a primitive matrix representation?

Question

Let $p(x)=x^6-13x^4-20x^3+x^2-x+2$ and $C$ be the companion matrix of $p(x)$.

How can I find a primitive matrix similar to $C$ ?

Is there a general method to transform the companion matrix with a Perron root into a primitive matrix?

Answer 1

Does every polynomial with a Perron root have a primitive matrix representation?

I found a counterexample, so in general the answer is no!

Let $p(x)=x^6-4x^4-5x^3+10x^2-12x+6$ and suppose A is a nonnegative matrix

similar to the companion matrix $C$ of $p(x)$. Then $Tr(A^k)\ge0$ for every natural number $k$, but

$Tr(C^4)=-8<0$. (Note that $A^k$ is similar to $C^k$.)

Hence $p(x)$ has no primitive matrix representation.

Answer 2

You do not ask the right question; in fact you show that the minimal polynomial of a Perron number (cf. below) is not necessarily the characteristic polynomial of a primitive integral matrix.

Let $\lambda>0$.

Def 1. A Perron number is a real algebraic integer exceeding $1$ and strictly greater than the modulus of all of its algebraic conjugates.

Def 2. $\lambda$ is a H-number iff there is an integer $k\geq 1$ s.t. $\lambda^k$ is a Perron number that has no positive real algebraic conjugate.

Prop 1. $\lambda$ is a H-number iff it is a root of a polynomial $z^n-\sum_{i=0}^{n-1}a_iz^i$ where the $a_i$ are non-negative integers and $a_0\not= 0$.

Prop 2. $\lambda$ is a Perron number iff it is the spectral radius of a primitive integral matrix.

Prop 3. $\lambda$ is a H-number iff it is the spectral radius of a primitive integral companion matrix.

Here your polynomial $p$ is irreducible; thus all matrices with characteristic polynomial $p$ are similar to $C_p$, the companion matrix of $p$. On the other hand, the maximal root $\lambda\approx 2.25$ of $p$ is a Perron number but not a H-number. According to Prop 2, $\lambda$ is the spectral radius of a primitive integral matrix $A$. According to your answer, $A$ has dimension $>6$. To construct such a matrix $A$, cf. the paper of Lind

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/3445712171510E3FF3A2C9543E995252/S0143385700002443a.pdf/entropies_of_topological_markov_shifts_and_a_related_class_of_algebraic_integers.pdf

Consider the irreducible polynomial $q=x^6-13x^4-20x^3-x^2-x-2$; its maximal root $\approx 2.78$ is a H-number (Prop 1). Let $C_q$ be the companion matrix of $q$; we can see that ${C_q}^8$ is a positive matrix.

You can also read the paper of Miss Bassino

https://ac.els-cdn.com/S0304397596001065/1-s2.0-S0304397596001065-main.pdf?_tid=17ed6aba-01f4-11e8-b347-00000aacb35d&acdnat=1516901032_a3c7f4ccb543e9873b2c267cf58e90c4