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Latent Roots of Tri-Diagonal Matrices

  • F. M. Arscott (a1)
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A considerable amount is known about the latent roots of matrices of the form

in the case when each cross-product of non-diagonal elements, aici-1, is positive. One forms the sequence of polynomials fr(λ) = |Lr−λI| for r = 1, 2, … n, and observes that

then it is easy to deduce that (i) the zeros of fn(λ) and fn_1(λ) interlace—that is, between two consecutive zeros of either polynomial lies precisely one zero of the other (ii) at the zeros of fn(λ) the values of fn-x(λ) are alternately positive and negative, (iii) all the zeros of fn(λ)— i.e. all the latent roots of Ln—are real and different.

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Edinburgh Mathematical Notes
  • ISSN: 0950-1843
  • EISSN: 2051-2031
  • URL: /core/journals/edinburgh-mathematical-notes
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