It is well known that on every $d$-dimensional torus there exists linear Anosov automorphisms. 


My question is the following:


Given $k< d$ does there exists a linear Anosov automorphism of $\mathbb{T}^d$ with exactly $k$ eigenvalues smaller than $1$? If true (which I expect), does there exists an \emph{irreducible} linear Anosov automorphism of $\mathbb{T}^d$ with exactly $k$ eigenvalues smaller than $1$? 


This can be phrased in terms of matrices with integer coeficients (please add the corresponding relevant tags) as:


Given $k< d$ does there exists a matrix in $SL(d,\mathbb{Z})$ such that all eigenvalues have modulus different from $1$ and $k$ of them are of modulus smaller than $1$? What about if the characteristic polynomial is irreducible over $\mathbb{Q}$?. 


Some relevant related information can be found in this paper (<http://arxiv.org/pdf/1009.2994v2.pdf>) where some results of W. Duke, Z. Rudnick, P. Sarnak as well as of Nevo and Sarnak are refered to. 



Eigenvalues for toral Anosov automorphisms
Given $k < d$, one can always construct a monic polynomial irreducible over $\mathbb Q$ with exactly $k$ roots less than 1 in modulus and $d-k$ roots greater than 1 in modulus. This follows from the general construction of algebraic units, namely, each group of units of an algebraic field contains a unit with a given $k$ -- see, e.g., [Borevich and Shafarevich]. 


Then you can simply take the companion matrix of such a polynomial. 


Or do you need an explicit construction?


