The smallest singular value anomaly: The reasons behind sharp anomaly

Abstract

Abstract Let A A be an arbitrary matrix in which the number of rows, m m , is considerably larger than the number of columns, n n . Let the submatrix A i , i = 1 , … , m {A}_{i},\hspace{0.33em}i=1,\ldots ,m , be composed from the first i i rows of A A , and let β i {\beta }_{i} denote the smallest singular value of A i {A}_{i} . Recently, we observed that the first part of this sequence, β 1 , … , β n {\beta }_{1},\ldots ,{\beta }_{n} , is descending, while the second part, β n , … , β m {\beta }_{n},\ldots ,{\beta }_{m} , is ascending. This property is called “the smallest singular value anomaly.” In this article, we expose another interesting feature of this sequence. It is shown that certain types of matrices possess the sharp anomaly phenomenon: First, when i i is considerably smaller than n n , the value of β i {\beta }_{i} decreases rather slowly. Then, as i i approaches n n from below, there is fast reduction in the value of β i {\beta }_{i} , making β n {\beta }_{n} much smaller than β 1 {\beta }_{1} . Yet, once i i passes n n , the situation is reversed and β i {\beta }_{i} increases rapidly. Finally, when i i moves away from n n , the rate of increase slows down. The article illustrates this behavior and explores its reasons. It is shown that the sharp anomaly phenomenon occurs in matrices with “scattering rows.”