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

Marko Lindner

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.

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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.
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.
• Marko Lindner
• Techn. Univ. Hamburg (TUHH), Institut Mathematik, D-21073 Hamburg, Germany
• Communicated by P.A. Cojuhari.
• Revised: 2018-04-04.
• Accepted: 2018-04-04.
• Published online: 2018-07-05.