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Could you tell me please, are there any analytical methods how to find eigenvalues of matrix such this one?

$$ \begin{pmatrix} a_1 & b_1 & 0 & 0 & 0 & \ldots & 0 \\ b_1 & a_2 & b_2 & 0 & 0 & \ldots & 0\\ 0 & b_2 & a_3 & b_3 & 0 & \ldots & 0 \\ 0&0 &b_3 & a_4&b_4 & \ldots & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & 0 & \ldots & a_{n-1} & b_{n-1}\\ 0 & 0 & 0& 0& \ldots & b_{n-1} & a_n \end{pmatrix} $$

I've found previously solutions for some special cases, but here all matrix elements are different and nonzero.

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    $\begingroup$ Any symmetric matrix can be brought to tridiagonal form through finitely many explicit steps. So if there were any explicit analytical solutions for the eigenvalues of a tridiagonal matrix, they would also apply to ALL symmetric matrices. Thus I think you're asking for too much! $\endgroup$
    – Igor Khavkine
    Commented Feb 22, 2020 at 21:45

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I am not sure what's the exact meaning of "analytic" in "analytic methods". If you expand $|A-\lambda I|$ in the last row or column twice, you obtain a "three term recurrency" for the characteristic polynomials. Polynomials satisfying this type of recurrencies have been studied VERY much, and they have many remarkable properties. They are called orthogonal polynomials.

The literature on these matrices and polynomials is really enormous. There are few cases which can be solved "explicitly".

See, for example, Gantmakher and Krein, Oscillation matrices and kernels and small vibrations of mechanical systems, AMS Chelsea Publishing, Providence, RI, 2002.

MR2743058 Simon, Barry Szegő's theorem and its descendants. Princeton University Press, Princeton, NJ, 2011.

N. I. Akhiezer, Classical moment problem, Hafner Publishing Co., New York 1965.

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  • $\begingroup$ Yes, the case in question is probably too general. Thanks for the answer. $\endgroup$
    – MightyPower
    Commented Feb 23, 2020 at 18:39
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Here is an elementary resource which gives details of a method along with nice examples.

http://homepage.divms.uiowa.edu/~atkinson/m171.dir/sec_9-4.pdf

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  • $\begingroup$ Thanks for the answer and for the resource. $\endgroup$
    – MightyPower
    Commented Feb 23, 2020 at 18:40

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