Date: Wednesday, February 20, 2013
Location: 4088 East Hall (3:10 PM to 4:00 PM)
Title: Critical Exponents
Abstract: Let $\mathcal{P}$ be a class of matrices, and let $A$ be an $m$-by-$n$ matrix in the class; consider some continuous powering, $A^{\{t\}}$. The critical exponent of $\mathcal{P}$, if it exists, with respect to the powering is the lowest power $g(\mathcal{P})$ such that for any matrix $B \in \mathcal{P}$ , $B^{\{t\}} \in \mathcal{P} \ \forall \ t > g(\mathcal{P})$. For powering relative to matrix multiplication in the traditional sense, hereafter referred to as \emph{conventional} multiplication, this means that $A^t$ is in the specified class for all $t > g_C(\mathcal{P})$. For Hadamard multiplication, similarly, $A^{(t)}$ is in the class $\forall \ t > g_H(\mathcal{P})$. We consider two questions for several classes $\mathcal{P}$ (including doubly nonnegative and totally positive): 1) does a critical exponent $g(\mathcal{P})$ exist? and 2) if so, what is it? For those where no exact result has been determined, lower and upper bounds are provided.
Speaker: Olivia Walch
Institution: University of Michigan
Event Organizer: Bobbie Wu boweiwu@umich.edu
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