Media type: E-Article Title: The max‐length‐vector line of best fit to a set of vector subspaces and an optimization problem over a set of hyperellipsoids Contributor: Bates, Daniel J.; Davis, Brent R.; Kirby, Michael; Marks, Justin; Peterson, Chris imprint: Wiley, 2015 Published in: Numerical Linear Algebra with Applications, 22 (2015) 3, Seite 453-464 Language: English DOI: 10.1002/nla.1965 ISSN: 1070-5325; 1099-1506 Keywords: Applied Mathematics ; Algebra and Number Theory Origination: Footnote: Description: <jats:title>Summary</jats:title><jats:p>Let <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/nla1965-math-0001.png" xlink:title="urn:x-wiley:nla:media:nla1965:nla1965-math-0001" /> be a collection of subspaces of a finite‐dimensional real vector space <jats:italic>V</jats:italic>. Let <jats:italic>L</jats:italic> denote a one‐dimensional subspace of <jats:italic>V</jats:italic>, and let <jats:italic>θ</jats:italic>(<jats:italic>L</jats:italic>,<jats:italic>V</jats:italic><jats:sub><jats:italic>i</jats:italic></jats:sub>) denote the principal angle between <jats:italic>L</jats:italic> and <jats:italic>V</jats:italic><jats:sub><jats:italic>i</jats:italic></jats:sub>. Motivated by a problem in data analysis, we seek an <jats:italic>L</jats:italic> that maximizes the function <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/nla1965-math-0002.png" xlink:title="urn:x-wiley:nla:media:nla1965:nla1965-math-0002" />. Conceptually, this is the line through the origin that best represents <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/nla1965-math-0003.png" xlink:title="urn:x-wiley:nla:media:nla1965:nla1965-math-0003" /> with respect to the criterion <jats:italic>F</jats:italic>(<jats:italic>L</jats:italic>). A reformulation shows that <jats:italic>L</jats:italic> is spanned by a vector <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/nla1965-math-0004.png" xlink:title="urn:x-wiley:nla:media:nla1965:nla1965-math-0004" />, which maximizes the function <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/nla1965-math-0005.png" xlink:title="urn:x-wiley:nla:media:nla1965:nla1965-math-0005" /> subject to the constraints <jats:italic>v</jats:italic><jats:sub><jats:italic>i</jats:italic></jats:sub>∈<jats:italic>V</jats:italic><jats:sub><jats:italic>i</jats:italic></jats:sub> and ||<jats:italic>v</jats:italic><jats:sub><jats:italic>i</jats:italic></jats:sub>||=1. In this setting, <jats:italic>v</jats:italic> is seen to be the longest vector that can be decomposed into unit vectors lying on prescribed hyperspheres. A closely related problem is to find the longest vector that can be decomposed into vectors lying on prescribed hyperellipsoids. Using Lagrange multipliers, the critical points of either problem can be cast as solutions of a multivariate eigenvalue problem. We employ homotopy continuation and numerical algebraic geometry to solve this problem and obtain the extremal decompositions. Copyright © 2015 John Wiley & Sons, Ltd.</jats:p>