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Let $n>m$, and let $A$ be an $(m\times n)$-matrix of full rank. Thenobviously the estimate $\|Ax\|\leq\|A\|\|x\|$ holds for the euclidean norm of$x$ and $Ax$ and the spectral norm as the assigned matrix norm. We study thesets of all $x$ for which, for fixed $\delta<1$, conversely$\|Ax\|\geq\delta\,\|A\|\|x\|$ holds. It turns out that these sets fill, in thehigh-dimensional case, almost the complete space once $\delta$ falls below abound that depends on the extremal singular values of $A$ and on the ratio ofthe dimensions. This effect has much to do with the random projection theorem,which plays an important role in the data sciences. As a byproduct, wecalculate the probabilities this theorem deals with exactly.

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