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1

Falseness of the finiteness property of the spectral subradius

Czornik A., Jurgaś P.

International Journal of Applied Mathematics and Computer Science

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2007

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Vol. 17, no 2

173-178

EN

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 it is not constructive because we do not show any explicit value of α that has described property. The problem of finding such values is still open.

2

O stabilności dyskretnych układów liniowych

Czornik A., Jurgaś P.

Zeszyty Naukowe. Automatyka / Politechnika Śląska

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2006

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z. 145

63-68

PL

Rozważamy tutaj skończone zbiory macierzy [sigma] = {A1, A2,…An} takie, że promień spektralny każdej macierzy takiego zbioru jest mniejszy niż 1. Następnie formułujemy twierdzenie pokazujące, że dla takiego zbioru możemy zawsze skonstruować zbiór [sigma]' = {A1(l1), A2(l2),…An(ln)} (l1, l2,…,ln) taki, że promień spektralny tego zbioru jest mniejszy niż l ([ro]( [sigma]') < l).

EN

We consider finite sets of matrices [sigma] = {A1, A2,…An} such that spectral radius of each matrix that belongs to such set is less than 1. Then we formulate the theorem that shows that for such set we can always construct the set [sigma]' = {A1(l1), A2(l2),…An(ln)} (l1, l2,…,ln) such that spectral radius o this set is less than 1 ([ro]( [sigma]') < l).

3

Some properties of the spectral radius of a set of matrices

Czornik A., Jurgaś P.

International Journal of Applied Mathematics and Computer Science

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2006

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Vol. 16, no 2

183-188

EN

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.