ACC Seminar: Growth in Products of Matrices

Refreshments will be served at 4:00 PM in the North Building, Room 316

ABSTRACT

The problems that we consider in this talk are as follows. Let A and B be 2x2 matrices (over reals). Let w(A, B) be a word of length n. After evaluating w(A, B) as a product of matrices, we get a 2x2 matrix, call it W. What is the largest (by the absolute value) possible entry of W, over all w(A, B) of length n, as a function of n? What is the expected absolute value of the largest (by the absolute value) entry in a random product of n matrices, where each matrix is A or B with probability 0.5? What is the Lyapunov exponent for a random matrix product like that? We give a partial answer to the first of these questions and an essentially complete answer to the second question. For the third question (the most difficult of the three), we offer a very simple method to produce an upper bound on the Lyapunov exponent in the case where all entries of the matrices A and B are nonnegative.

BIOGRAPHY

Vladimir Shpilrain is Professor of Mathematics at the City College of New York. He received his PhD from the Lomonosov Moscow State University under the direction of Alfred Lvovich Shmelkin. Professor Shpilrain's research interests include combinatorial and computational group theory, information security, and complexity of algorithms.

Attendance: This is a technical talk open to all.
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Additional information is available at https://web.stevens.edu/algebraic/.

Zoom Link:

https://stevens.zoom.us/j/93228680142 (Password: ACC)