Abstract
Abstract A recent work derived expressions for the induced p-norm of a special class of circulant matrices A(n, a, b) ∈ ℝ n × n , with the diagonal entries equal to a ∈ ℝ and the off-diagonal entries equal to b ≥ 0. We provide shorter proofs for all the results therein using Fourier analysis. The key observation is that a circulant matrix is diagonalized by a DFT matrix. The results comprise an exact expression for ǁAǁ p , 1 ≤ p ≤ ∞, where A = A(n, a, b), a ≥ 0 and for ǁAǁ2 where A = A(n, −a, b), a ≥ 0; for the other p-norms of A(n, −a, b), 2 < p < ∞, upper and lower bounds are derived.
Highlights
The key observation is that a circulant matrix is diagonalized by a Discrete Fourier Transform (DFT) matrix
The results comprise an exact expression for A p, ≤ p ≤ ∞, where A = A(n, a, b), a ≥ and for A where A = A(n, −a, b), a ≥ ; for the other p-norms of A(n, −a, b)
Circulant matrices arise in many applications ranging from wireless communication [10] to cryptography [7] to solving di erential equations [11]
Summary
Circulant matrices arise in many applications ranging from wireless communication [10] to cryptography [7] to solving di erential equations [11] (see [2] and the references therein for historical context and more recent theoretical studies on circulant matrices). Abstract: A recent work derived expressions for the induced p-norm of a special class of circulant matrices A(n, a, b) ∈ Rn×n, with the diagonal entries equal to a ∈ R and the o -diagonal entries equal to b ≥ . We provide shorter proofs for all the results therein using Fourier analysis.
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