Let n>m and A be an (m\times n)-matrix of full rank. Then obviously the estimate \|Ax\|\leq\|A\|\|x\| holds for the euclidean norm of Ax. We study in this paper the sets of all x for which conversely \|Ax\|\geq\delta\,\|A\|\|x\| holds for some \delta<1. It turns out that these sets fill in the high-dimensional case almost the complete space once \delta 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 sizeof these dimensions .
Let n>m and A be an (m\times n)-matrix of full rank. Then obviously the estimate \|Ax\|\leq\|A\|\|x\| holds for the euclidean norm of Ax. We study in this paper the sets of all x for which conversely \|Ax\|\geq\delta\,\|A\|\|x\| holds for some \delta<1. It turns out that these sets fill in the high-dimensional case almost the complete space once \delta 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 their size .
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the sizeof these dimensions
their size