Hello,


Let $A\_n = (a\_{k-j};\;k,j = 0,1,\ldots,n-1)$ be a sequence of $n\times n$ Toeplitz matrices, with eigenvalues $(\lambda\_{n,i};\;i = 0,1,\ldots,n-1)$. 


If $A\_n$ were a sequence of Hermitian Toeplitz matrices, and if $\sum\_k|a\_k|<\infty$, then Szego theorem states that for any continues function $F(\cdot)$ on $[\alpha,\beta]$ we have
$$
\lim\_{n\to\infty}\frac{1}{n}\sum\_{k=0}^{n-1}F(\lambda\_{n,k}) = \frac{1}{2\pi}\int\_0^{2\pi}F(f(\xi))d\xi
$$  

where
$$
f(\xi) = \sum\_{k = -\infty}^{\infty}a\_ke^{ik\xi}
$$
and $\alpha = \text{ess}\inf f$ and $\beta = \text{ess} \sup f$. 


If however $A\_n$ are not-Hermitian, then the above hold only for polynomial functions, namely
$$
\lim\_{n\to\infty}\frac{1}{n}\sum\_{k=0}^{n-1}p(\lambda\_{n,k}) = \frac{1}{2\pi}\int\_0^{2\pi}p(f(\xi))d\xi
$$
where $p(\cdot)$ is some polynomial function. 


My question: is there any result regard the asymptotic behavior of the modulus of the eigenvalues for the non-Hermitian case, namely
$$
\lim\_{n\to\infty}\frac{1}{n}\sum\_{k=0}^{n-1}|\lambda\_{n,k}| = ?.
$$




---


EDITION: due to the thankful Alexandersson comment, I add the factor $1/n$... 



Asymptotic Behavior of Non-Analytic Function of the Eigenvalues   
To create a partial answer, 
eigenvalues of banded Toeplitz matrices accumulate on some real algebraic curves in $\mathbb{C}$, (this has been proved by Schmidt and Spitzer around 1970.).
Now, if we also attach a point-mass at each eigenvalue, (for fixed matrix size) with equal mass at each point, and with total mass one, we get a probability measure.
These measures converge in a certain sense to some limit measure,
see for example the book by [Bender and Böttcher](http://rads.stackoverflow.com/amzn/click/0898715997) .


Now, the sum $c\_n = \frac{1}{n} \sum\_{k=1}^n |\lambda\_{n,k}|$ can then be interpreted 
as the ~~center of mass~~ mean distance to 0 of all the eigenvalues from matrix of size $n \times n.$
Say that the associated point measure is called $\mu\_n$ (mass $1/n$ at each eigenvalue),
then we know that $\mu\_n \to \mu$ for some $\mu$ in some sense.


Note that $c\_n = \int\_{\mathbb{C}} |z| d\mu\_n(z)$ and then what you are looking for is $c = \int\_{\mathbb{C}} |z| d\mu(z)$.


As an explicit example, taking your matrix to be tridiagonal, will give rise to characteristic polynomials which are also a family of orthogonal polynomials, w.r.t some measure (the limit of point measures, actually), see e.g. [this paper i just googled](http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CDkQFjAA&url=http%3A%2F%2Feuler.us.es%2F~renato%2Fpapers%2Fchain-revised-1.pdf&ei=iTrpUIi5MaOH4ASojYDADQ&usg=AFQjCNGMKq-EYEq5rqZeXhVcv4eqrHj2Lw&bvm=bv.1355534169,d.bGE&cad=rja).


