A measure concentration effect for matrices of high, higher, and even higher dimension

10/26/2020

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by   Harry Yserentant, et al.

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Let n>m and A be an (m× n)-matrix of full rank. Then obviously the estimate Ax≤Ax holds for the euclidean norm of Ax. We study in this paper the sets of all x for which conversely Ax≥δ Ax holds for some δ<1. It turns out that these sets fill in the high-dimensional case almost the complete space once δ falls below a certain bound that depends only on the condition number of A and on the ratio of the dimensions m and n, but not on the size of these dimensions.