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Geometric computation of the numerical radius of a matrix

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Abstract

We use methods of geometric computing combined with hermitean matrix eigenvalue/eigenvector evaluations to find the numerical radius w(A) of a real or complex square matrix A simply, quickly, and accurately. The numerical radius w(A) is defined as the maximal distance of points in the field of values \(F(A) = \{ x^* A x \mid \|x\|_2 = 1 \}\) from zero in ℂ. Its value is an indicator of the transient behavior of the discrete dynamical system f k + 1 = Af k . We describe and test a MATLAB code for solving this optimization problem that can have multiple and even infinitely many solutions with maximal distance.

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  1. Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36849-5310, USA

    Frank Uhlig

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  1. Frank Uhlig
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Correspondence to Frank Uhlig.

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Uhlig, F. Geometric computation of the numerical radius of a matrix. Numer Algor 52, 335–353 (2009). https://doi.org/10.1007/s11075-009-9276-1

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Mathematics Subject Classifications (2000)