ABSTRACT
There are several reasons to study the circulant matrix and its variants. For example, the large dimensional non-random Toeplitz matrix and the corresponding Toeplitz operator are well-studied objects in mathematics. The non-random circulant matrix plays a crucial role in these studies. See, for example, Grenander and Szegő (1984) and Gray (2006). The eigenvalues of the circulant matrix also arise crucially in time series analysis. For instance, the periodogram of a sequence {al }l≥0 is defined as https://www.w3.org/1998/Math/MathML"> n − 1 | ∑ l = 0 n − 1 a l e 2 π i j l / n | 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429435508/eeca0458-37b4-425a-950e-71e45ea131f7/content/eq23.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , https://www.w3.org/1998/Math/MathML"> − ⌊ n − 1 2 ⌋ ≤ j ≤ ⌊ n − 1 2 ⌋ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429435508/eeca0458-37b4-425a-950e-71e45ea131f7/content/eq24.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and it is a straightforward function of the eigenvalues of a suitable circulant matrix. The properties of the periodogram are fundamental in the spectral analysis of time series. See for instance Fan and Yao (2003). The maximum of the periodogram, in particular, has been studied in Davis and Mikosch (1999). The k-circulant matrices and their block versions arise in areas such as multi-level supersaturated design of experiment (Georgiou and Koukouvinos (2006)), spectra of De Bruijn graphs (Strok (1992)) and (0,1)-matrix solutions to Am = Jn (Wu et al. (2002)). See also Davis (1979) and Pollock (2002).
