×

Circulant matrices: norm, powers, and positivity. (English) Zbl 1406.15024

Let \(n\geq 2\), \(\mathbf x=(x_0,\dots,x_{n-1})\in\mathbb R^n\), let \(\mathbf{C_x}\) be the corresponding circulant matrix, and let \(\|\cdot\|\) denote the spectral norm. The reviewer et al., [J. K. Merikoski et al., Spec. Matrices 6, 23–36 (2018; Zbl 1392.15044)] proved that if (a) \(\mathbf{B_x}=\mathbf{C}^T_{\mathbf x}\mathbf{C_x}\geq\mathbf 0\) (entrywise), then (b) \(\|\mathbf{C_x}\|=|x_0+\dots+x_{n-1}|\). The present author improves this by showing (Theorem 2.1a) that if (a’) \(\mathbf{B}^m_{\mathbf x}\geq\mathbf 0\) for some \(m\in\mathbb N\), then (b) holds. He also notes (Example 2.3) that (a’) is not necessary for (b).
Let \[ c(t)=x_0+x_1t+\dots+x_{n-1}t^{n-1},\quad T_n=\{1,\omega,\dots,\omega^{n-1}\}, \quad\omega=\exp{\frac{2\pi{\mathrm i}}{n}}. \] Equivalent to (b) is that \(\max\{|c(t)|\mid t\in T_n\}\) is attained if \(t=1\). This follows from the fact that \(\mathbf C_x\) is unitarily similar to \(\mathrm{diag}\,(c(1),c(\omega),\dots,c(\omega^{n-1}))\) via the Fourier matrix \(\mathbf{F}=(f_{ij})\), \(f_{ij}=\omega^{(i-1)(j-1)}/\sqrt{n}\). Let (b’) denote the stronger condition: this maximum is attained if and only if \(t=1\). The author proves (Theorem 2.2 and Corollary 2.4) that (b’) holds if and only if \(\mathbf{B}^m_{\mathbf x}>\mathbf 0\) for some \(m\in\mathbb N\) or, equivalently, for all sufficiently large values of \(m\).
Finally, the author extends his results to complex circulant matrices.
Reviewer’s remark: In the bibliography, [2] actually refers to the preprint.

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15B05 Toeplitz, Cauchy, and related matrices
15B48 Positive matrices and their generalizations; cones of matrices

Citations:

Zbl 1392.15044
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] P.J. Davis, Circulant Matrices, Wiley, 1979.
[2] J.K. Merikoski, P. Haukkanen, M. Mattila, T. Tossavainen, The spectral norm of a Horadam circulant matrix, JP Journal of Algebra, Number Theory and Applications, to appear. · Zbl 1392.15044
[3] J.K. Merikoski, P. Haukkanen, M. Mattila, T. Tossavainen, On the spectral and Frobenius norm of a generalized Fibonacci r-circulant matrix, Special Matrices 6 (2018), 23–36. Circulant matrices: norm, powers, and positivity857 Marko Lindner lindner@tuhh.de Techn. Univ. Hamburg (TUHH) Institut Mathematik D-21073 Hamburg, Germany Received: March 28, 2018. Revised: April 4, 2018. Accepted: April 4, 2018. · Zbl 1392.15044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.