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<p>Chi-Square distributions (or lack of it) and independence (or non-independence) of quadratic forms composed of independent standard normal variables form the core of all linear model theory which practically sits on the base made up of idempotent matrices. After a brief review, a very useful characterization of the class of all <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p\times p\)</EquationSource> </InlineEquation> idempotent matrices of rank <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(m(&lt;p)\)</EquationSource> </InlineEquation> is obtained with <i>p</i>,&#xa0;<i>m</i> fixed. Having armed with that, a number of explicitly complicated quadratic forms are exhibited including some that may appear rather daunting, and examined their distributions and independence. Such non-trivial constructions also show how a large Chi-Square statistic may be fruitfully split into a sum (or a linear function) of smaller Chi-Squares through explicit examples. The paper ends with selected but simple-minded <i>exploratory data analysis</i> (EDA) and some brief concluding thoughts.</p>

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On Revisiting Idempotent Matrices and Chi-Square Distributions of Quadratic Forms

  • 期刊论文

  • 发表日期：

    2026年5月28日

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  • Nitis Mukhopadhyay

摘要

Chi-Square distributions (or lack of it) and independence (or non-independence) of quadratic forms composed of independent standard normal variables form the core of all linear model theory which practically sits on the base made up of idempotent matrices. After a brief review, a very useful characterization of the class of all \(p\times p\) idempotent matrices of rank \(m(<p)\) is obtained with p, m fixed. Having armed with that, a number of explicitly complicated quadratic forms are exhibited including some that may appear rather daunting, and examined their distributions and independence. Such non-trivial constructions also show how a large Chi-Square statistic may be fruitfully split into a sum (or a linear function) of smaller Chi-Squares through explicit examples. The paper ends with selected but simple-minded exploratory data analysis (EDA) and some brief concluding thoughts.