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Open lecture “Joint spectral characteristics of matrices". Vladimir Protasov

description

To an arbitrary finite family of n x n matrices we assoiate its joint spectral characteristics which are exponents of the asymptotig growth of their long products (with no ordering and with repetitions permitted).

For one matrix, all its joint spectral characteristics coincide with its usulal spectral radius, while already for two matrices we obtain a nontrivial theory.  Joint spectral characteristics originated in 1960 with J.-K.Rota and G.Strang (the joint spectral radius) and with H.Furstenberg and H.Kesten (the Lyapunov exponent).  Then other characteristics appeared. Those characteristics have remarkable applications, including engineering ones.

They are also known by the difficlty of their computation, even for low dimensions. Nevertheless, recent methods can efficiently approximate the joint spectral characteristics and even find their exact values.  Each of those methods has a serious theoretical framework. We demonstrate and discuss  some of them and formulate several open problems.

Bio

Vladimir Protasov is a Doctor of Sciences (2006),  Professor of the Russian Academy of Sciences (2015), a corresponding member  of the Russian Academy of Sciences (2016). Also, he is a professor at  Moscow State University, at National Research University Higher School of Economics (Russia), and at University of L'Aquila (Italy). Dr. Protasov wrote 3 monographies and more than 80 journal papers. Also he published several books and papers in elementary mathematics for high school students.  His main research areas are functional analysis, wavelets, optimization, numerical analysis, algorithms, linear algebra and matrix analysis, geometry.  Five PhD thesys and 13 masters degree diplomas have been defended under his supervision.  Dr. Protasov has held visiting positions in universities and research centers of USA, France, Netherlands, Belgium, Italy, China, Turkey, and Hong Kong.

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