Opuscula Math. 38, no. 6 (2018), 849-857
https://doi.org/10.7494/OpMath.2018.38.6.849
Opuscula Mathematica
Circulant matrices: norm, powers, and positivity
Abstract. In their recent paper "The spectral norm of a Horadam circulant matrix", Merikoski, Haukkanen, Mattila and Tossavainen study under which conditions the spectral norm of a general real circulant matrix \({\bf C}\) equals the modulus of its row/column sum. We improve on their sufficient condition until we have a necessary one. Our results connect the above problem to positivity of sufficiently high powers of the matrix \({\bf C^\top C}\). We then generalize the result to complex circulant matrices.
Keywords: spectral norm, circulant matrix, eventually positive semigroups.
Mathematics Subject Classification: 15A60, 15B05, 15B48.
- P.J. Davis, Circulant Matrices, Wiley, 1979.
- 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.
- 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.
- Marko Lindner
- Techn. Univ. Hamburg (TUHH), Institut Mathematik, D-21073 Hamburg, Germany
- Communicated by P.A. Cojuhari.
- Received: 2018-03-28.
- Revised: 2018-04-04.
- Accepted: 2018-04-04.
- Published online: 2018-07-05.
