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I would like to ask for a reference on the problem of computing the eigenvalues/eigenvectors of tridiagonal matrices (not necessarily with constant diagonals).

I have seen authors use continued fractions and generating functions. However, I have thus far been unable to really grasp the foundations of this idea. From what I can see, the idea is really to reduce it to a difference equations. Then, perhaps, my request is for a good book on difference equations. Moreover, is there any technique which is really of a broad scope; ie applicable to a broad range of problems.

Thank you all in advance,

Gabieel

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    A few notes: if the products of the corresponding subdiagonal and superdiagonal elements are positive, you can, through a similarity transformation, always reduce your tridiagonal eigenproblem to a *symmetric* tridiagonal eigenproblem, which is very much well studied. The unsymmetric eigenproblems that do not yield to this treatment are said to be a difficult case for the QR algorithm and related methods.2012-08-20
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    See Horn and Johnson: Topics in Matrix Analysis. This book is reference in matrix analysis.2012-11-15
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    @Elias Great. Thank you very much.2012-11-17
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    @Elias Please consider converting your comment into an answer, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2013-06-22

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