Spectral radius of sets of matrices

Mohsen Maesumi

doi:10.1117/12.188808

11 October 1994 Spectral radius of sets of matrices

Mohsen Maesumi

Proceedings Volume 2303, Wavelet Applications in Signal and Image Processing II; (1994) https://doi.org/10.1117/12.188808
Event: SPIE's 1994 International Symposium on Optics, Imaging, and Instrumentation, 1994, San Diego, CA, United States

Abstract

Spectral radius of sets of matrices is a fundamental concept in studying the regularity of compactly supported wavelets. Here we review the basic properties of spectral radius and describe how to increase the efficiency of estimation of a lower bound for it. Spectral radius of sets of matrices can be defined by generalizing appropriate definitions of spectral radius of a single matrix. One definition, referred to as generalized spectral radius, is constructed as follows. Let (Sigma) be a collection of m square matrices of same size. Suppose L_n((Sigma) ) is the set of products of length n of elements (Sigma) . Define p_n((Sigma) ) equals max_A(epsilon L_n [p(A)]^1/n where p(A) is the usual spectral radius of a matrix. Then the generalized spectral radius of (Sigma) is p((Sigma) ) equals lim sup_{nyields(infinity} )p_n((Sigma) ). The standard method for estimating p((Sigma) ), through p_n((Sigma) ), involves mⁿ matrix calculations, one per each element of L_n((Sigma) ). We will describe a method which reduces this cost to mⁿ/n or less.

Citation Download Citation

Mohsen Maesumi "Spectral radius of sets of matrices", Proc. SPIE 2303, Wavelet Applications in Signal and Image Processing II, (11 October 1994); https://doi.org/10.1117/12.188808

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