Latent Roots of Tri-Diagonal Matrices
A considerable amount is known about the latent roots of matrices of the formin 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 thatthen 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.
1904 ◽
Vol 24
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pp. 233-239
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1878 ◽
Vol 9
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pp. 332-333
1906 ◽
Vol 25
(2)
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pp. 806-812
1893 ◽
Vol 19
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pp. 15-19
1906 ◽
Vol 25
(2)
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pp. 903-907
1939 ◽
Vol 6
(2)
◽
pp. 75-77