Toeplitz and Circulant Matrices

Toeplitz and Circulant Matrices: A Review, by R. M. Gray. A very old (1971, revised 1977, 1993, 1997, 1998, 2000, 2001, 2002, 2005, 2006) but still occasionally useful tutorial on Toeplitz and circulant matrices.

The report was revised with the help of two very thorough reviewers and is being published both online and as a paperback book by NOW publishers. The official citation to the published version is
R. M. Gray, "Toeplitz and Circulant Matrices: A review"
Foundations and Trends in Communications and Information Theory,
Vol 2, Issue 3, pp 155-239, 2006.
Journal reprint

Note: The typos found and noted below are corrected in the first pdf, but not in the second.

A printed and bound version of the paperback book is available at a 35% discount from Now Publishers.

This can be obtained by entering the promotional code CITMC06 on the order form at now publishers.
You will then pay only $28.00 including postage.
(The Website is due to be activated as soon as the book is available.)

Typographical errors:

• On p. 33, the first equation: y^{(m)} = \frac{1}{\sqrt{n}}\( 1, e^{-2\pi i m/n}, \ldots, e^{-2\pi i (n - 1)/n} \). The last exponent should be $e^{-2\pi i m(n - 1)/n}$?
• On p. (62) the explanation of (5.1) is garbled. It published latex is
  \begin{equation}\label{4.40}
  \mid (x,y,z)\defn
  \begin{cases}
  z & \text{$y \ge z$}\\
  y & \text{$x \le y \le z$}\\
  x & \text{$y\le x$}\\z & \text{$y \ge z$}\\
  \end{cases}
  \end{equation}
  $x < z$.
  \begin{equation}\label{4.40}
  \mid (x,y,z)\defn
  \begin{cases}
  This function can be thought of as having input
  $y$ and thresholds $z$ and $X$ and it puts out $y$ if $y$ is between
  $z$ and $x$, $z$ if $y$ is smaller than $z$, and $x$ if $y$ is greater
  than $x$.
  It should say
  \begin{equation}\label{4.40}
  \mid (x,y,z)\defn
  \begin{cases}
  z & \text{$y \ge z$}\\
  y & \text{$x \le y \le z$}\\
  x & \text{$y\le x$}\\
  \end{cases}
  \end{equation}
  This function can be thought of as having input
  $y$ and thresholds $z$ and $x$ and it puts out $y$ if $y$ is between
  $z$ and $x$, $z$ if $y$ is greater than $z$, and $x$ if $y$ is less
  than $x$.
  

Comments and corrections are welcome to rmgray@stanford.edu.