A Frequently Encountered Determinant

Alexander A. Roytvarf

A chapter in Thinking in Problems, 2013, pp 37-41 from Springer

Abstract: Abstract The following matrix appears in various problems connected with equidistants and envelopes in analysis, geometry, calculation of variations, and mathematical physics: $$ A=\left( {\begin{array}{*{20}{c}} {1+{a_1}^2} & {{a_1}{a_2}} & {\ldots } & {{a_1}{a_n}} \\ {{a_2}{a_1}} & {1+{a_2}^2} & {\ldots } & {{a_2}{a_n}} \\ . & . & {\ldots } & . \\ {{a_n}{a_1}} & . & {\ldots } & {1+{a_n}^2} \\ \end{array}} \right) . $$ A calculation of determinants of this matrix and matrices of a more general form is a nice exercise in linear algebra that does not require advanced knowledge on the part of readers (however, readers possessing such knowledge who are interested in applications will find in this chapter a typical application example and a brief follow-up discussion).

Keywords: Implicit Function Theorem; Morse Theory; Eikonal Equation; Advanced Knowledge; Hamiltonian Mechanic (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/978-0-8176-8406-8_4

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