The smallest singular value anomaly: The reasons behind sharp anomaly

Abstract

Let AA be an arbitrary matrix in which the number of rows, mm, is considerably larger than the number of columns, nn. Let the submatrix Ai,i=1,…,m{A}_{i},\hspace{0.33em}i=1,\ldots ,m, be composed from the first ii rows of AA, and let βi{\beta }_{i} denote the smallest singular value of Ai{A}_{i}. Recently, we observed that the first part of this sequence, β1,…,βn{\beta }_{1},\ldots ,{\beta }_{n}, is descending, while the second part, βn,…,βm{\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 ii is considerably smaller than nn, the value of βi{\beta }_{i} decreases rather slowly. Then, as ii approaches nn from below, there is fast reduction in the value of βi{\beta }_{i}, making βn{\beta }_{n} much smaller than β1{\beta }_{1}. Yet, once ii passes nn, the situation is reversed and βi{\beta }_{i} increases rapidly. Finally, when ii moves away from nn, 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.”

Keywords

• the smallest singular value anomaly
• the condition number anomaly
• iterations anomaly
• the sharp anomaly phenomenon
• the scattering rows property
• 65f15
• 65f25
• 65f50