On a devil’s staircase associated to the joint spectral radii of a family of pairs of matrices

Ian D. Morris; Nikita Sidorov

Displaying similar documents to “On a devil’s staircase associated to the joint spectral radii of a family of pairs of matrices”

Some properties of the spectral radius of a set of matrices

Adam Czornik, Piotr Jurgas (2006)

International Journal of Applied Mathematics and Computer Science

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In this paper we show new formulas for the spectral radius and the spectral subradius of a set of matrices. The advantage of our results is that we express the spectral radius of any set of matrices by the spectral radius of a set of symmetric positive definite matrices. In particular, in one of our formulas the spectral radius is expressed by singular eigenvalues of matrices, whereas in the existing results it is expressed by eigenvalues.

Vandermonde Matrices on the Circle: Spectral Properties and Conditioning.

W. Gautschi, A. Córdova, ... (1990)

Numerische Mathematik

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On the joint spectral radius

Vladimír Müller (1997)

Annales Polonici Mathematici

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We prove the $_{p}$ -spectral radius formula for n-tuples of commuting Banach algebra elements

Absolute continuity for Jacobi matrices with power-like weights

Wojciech Motyka (2007)

Colloquium Mathematicae

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This work deals with a class of Jacobi matrices with power-like weights. The main theme is spectral analysis of matrices with zero diagonal and weights $λ ₙ : = n^{α} (1 + Δ ₙ)$ where α ∈ (0,1]. Asymptotic formulas for generalized eigenvectors are given and absolute continuity of the matrices considered is proved. The last section is devoted to spectral analysis of Jacobi matrices with qₙ = n + 1 + (-1)ⁿ and $λ ₙ = \sqrt (q_{n - 1} q ₙ)$ .

Spectral Methods with Sparse Matrices.

Wilhelm Heinrichs (1989/90)

Numerische Mathematik

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On the zero set of the Kobayashi-Royden pseudometric of the spectral unit ball

Nikolai Nikolov, Pascal J. Thomas (2008)

Annales Polonici Mathematici

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Given A∈ Ωₙ, the n²-dimensional spectral unit ball, we show that if B is an n×n complex matrix, then B is a “generalized” tangent vector at A to an entire curve in Ωₙ if and only if B is in the tangent cone $C_{A}$ to the isospectral variety at A. In the case of Ω₃, the zero set of the Kobayashi-Royden pseudometric is completely described.

Convergence of series of dilated functions and spectral norms of GCD matrices

Christoph Aistleitner, István Berkes, Kristian Seip, Michel Weber (2015)

Acta Arithmetica

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We establish a connection between the L² norm of sums of dilated functions whose jth Fourier coefficients are $(j^{- α})$ for some α ∈ (1/2,1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L² and for the almost everywhere convergence of series of dilated functions.

On some spectral characteristics of multiparameter polynomial matrices.

Khazanov, V.B. (2004)

Zapiski Nauchnykh Seminarov POMI

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The Direct and Inverse Spectral Problems for some Banded Matrices

Zagorodnyuk, S. M. (2011)

Serdica Mathematical Journal

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2000 Mathematics Subject Classification: 15A29. In this paper we introduced a notion of the generalized spectral function for a matrix J = (gk,l)k,l = 0 Ґ, gk,l О C, such that gk,l = 0, if |k-l | > N; gk,k+N = 1, and gk,k-N № 0. Here N is a fixed positive integer. The direct and inverse spectral problems for such matrices are stated and solved. An integral representation for the generalized spectral function is obtained.

Spectral radius preservers of products of monnegative matrices.

Clark, Sean, Li, Chi-Kwong, Rodman, Leiba (2008)

Banach Journal of Mathematical Analysis [electronic only]

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Spectral variation bounds for diagonalisable matrices.

R. Bhatia, L. Elsner, G.M. Krause (1997)

Aequationes mathematicae

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Falseness of the finiteness property of the spectral subradius

Adam Czornik, Piotr Jurgas (2007)

International Journal of Applied Mathematics and Computer Science

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We prove that there exist infinitely may values of the real parameter α for which the exact value of the spectral subradius of the set of two matrices (one matrix with ones above and on the diagonal and zeros elsewhere, and one matrix with α below and on the diagonal and zeros elsewhere, both matrices having two rows and two columns) cannot be calculated in a finite number of steps. Our proof uses only elementary facts from the theory of formal languages and from linear algebra, but...

Quotient of spectral radius, (signless) Laplacian spectral radius and clique number of graphs

Kinkar Ch. Das, Muhuo Liu (2016)

Czechoslovak Mathematical Journal

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In this paper, the upper and lower bounds for the quotient of spectral radius (Laplacian spectral radius, signless Laplacian spectral radius) and the clique number together with the corresponding extremal graphs in the class of connected graphs with $n$ vertices and clique number $ω$ $(2 \leq ω \leq n)$ are determined. As a consequence of our results, two conjectures given in Aouchiche (2006) and Hansen (2010) are proved.

Formulae for joint spectral radii of sets of operators

Victor S. Shulman, Yuriĭ V. Turovskii (2002)

Studia Mathematica

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The formula $ϱ (M) = m a x ϱ_{χ} (M), r (M)$ is proved for precompact sets M of weakly compact operators on a Banach space. Here ϱ(M) is the joint spectral radius (the Rota-Strang radius), $ϱ_{χ} (M)$ is the Hausdorff spectral radius (connected with the Hausdorff measure of noncompactness) and r(M) is the Berger-Wang radius.

On the joint spectral radius of commuting matrices

Rajendra Bhatia, Tirthankar Вhattacharyya (1995)

Studia Mathematica

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For a commuting n-tuple of matrices we introduce the notion of a joint spectral radius with respect to the p-norm and prove a spectral radius formula.