The smallest singular value anomaly: The reasons behind sharp anomaly

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>A</m:mi> </m:math> A be an arbitrary matrix in which the number of rows, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>m</m:mi> </m:math> m , is considerably larger than the number of columns, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> </m:math> n . Let the submatrix <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mspace width="0.33em" /> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo form="prefix">,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>m</m:mi> </m:math> {A}ᵢ,{0.33em}i=1,… ,m , be composed from the first <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>i</m:mi> </m:math> i rows of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>A</m:mi> </m:math> A , and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>β</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:math> {β }ᵢ denote the smallest singular value of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:math> {A}ᵢ . Recently, we observed that the first part of this sequence, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>β</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo form="prefix">,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>β</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> {β }₁,… ,{β }ₙ , is descending, while the second part, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>β</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo form="prefix">,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>β</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msub> </m:math> {β }ₙ,… ,{β }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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>i</m:mi> </m:math> i is considerably smaller than <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> </m:math> n , the value of <m:math xmlns:m="http://www.w3.org/1998/Math/Ma