A Frequently Encountered Determinant

  • Alexander A. Roytvarf


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).


Implicit Function Theorem Morse Theory Eikonal Equation Advanced Knowledge Hamiltonian Mechanic 
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© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  • Alexander A. Roytvarf
    • 1
  1. 1.Rishon LeZionIsrael

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