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On the distribution of a scaled condition number

1992/01/01 by Alan Edelman · 3 citations
Mathematics · Computer Science · #Random Matrices and Applications #Matrix Theory and Algorithms #Advanced Combinatorial Mathematics #Algorithm #Computer science

paper · doi:10.1090/s0025-5718-1992-1106966-2

openalex publication_date 1992/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/21

Abstract

In this note, we give the exact distribution of a scaled condition number used by Demmel to model the probability that matrix inversion is difficult. Specifically, consider a random matrix <italic>A</italic> and the scaled condition number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa Subscript upper D Baseline left-parenthesis upper A right-parenthesis equals double-vertical-bar upper A double-vertical-bar Subscript upper F Baseline bullet double-vertical-bar upper A Superscript negative 1 Baseline double-vertical-bar"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi> κ </mml:mi> <mml:mi>D</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mi>A</mml:mi> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mi>F</mml:mi> </mml:msub> </mml:mrow> <mml:mo> ∙ </mml:mo> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">κ D(A) = ‖ A ‖F \bullet ‖ A - 1 ‖</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Demmel provided bounds for the condition number distribution when <italic>A</italic> has real or complex normally distributed elements. Here, we give the exact formula.

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