2
$\begingroup$

I keep encountering ostensibly very different branches of mathematics, only to have eigenvalues show up in each one. Is there a single book out there that presents a deep, unified account of the eigenvalue across different fields of math?

  • 3
    Eigenvalues are strictly linear algebraic term, as far as I am aware. Rather, it's linear algebra that is prevalent across many branches of mathematics. Or do you have an example of non-linear-algebraic meaning of "eigenvalue"?2012-08-14
  • 0
    I don't think there exists a single book that contains all topics in math. The same goes for eigenvalues.2012-08-14
  • 0
    Maybe [wikipedia](http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors) is the best you can get.2012-08-14
  • 2
    A study of abstract linear algebra (ie, vector spaces and linear operators, not matrices and $\mathbb{R}^n$) might provide the closest thing to an answer to this question. It can sometimes seem like the "eigenvalues" people mention in varied contexts are different from abstract linear algebra eigenvalues (e.g. because no linear operator or vector space is mentioned, just a physical system, or a differential equation). But with effort it is often possible to identify an operator, specific to each applied context, whose linear algebra eigenvalues are the "eigenvalues" under discussion.2012-08-14

1 Answers 1

1

There is exsit a book "Wilkinson - The Algebraic Eigenvalue Problem" I didn't read it.

Actually this is central part of all linear algebra. And it's miracle that we can solve it via finding all eigenvalues.

In general there are several algorithms:

https://www-old.math.gatech.edu/academic/courses/core/math2601/Web-notes/5num.pdf

http://cstl-csm.semo.edu/jwojdylo/MA345/Chapter6/qrmethod/qrmethod.pdf

But they don't solve "full spectral problem". But looks that John Francis and Vera Kublanovskaya solved it. (Even I'm familiar only with basic QR algorithm which works only with real eigenvalues)

https://en.wikipedia.org/wiki/Vera_Kublanovskaya

https://www.quora.com/What-are-the-top-10-algorithms-of-the-20th-century