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I found a book in the library about antieigenvalue analysis and it is possibly the most unreadable piece of literature I have ever made an effort to understand. Unfortunately, every other resource I try inevitably takes you back to the same author.

I should apologize for a lack of greater research effort, but beyond the line on the wikipedia page,

The antieigenvectors $x$ are the vectors most turned by a matrix or operator $A$

I can't make heads or tails of anything else.

Could someone explain why (or why not) this topic is useful or interesting?

I've previously read about topics such as fractional calculus, harmonic analysis, non-standard analysis, product integration, quantum probability, higher order fourier analysis and other topics by dusting off a rarely read book off a shelf. I've always found something cool or interesting.

I enjoy linear algebra quite a lot. The name antieigenvalue is very enticing to me. I really want to think that this topic is going to be neat. What is in antieigenvalue analysis that should excite me?

JessicaK
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  • I'm intrigued. Does the author at least give a definition of what he means by "turning angle"? Is there an index to look through? – Ben Grossmann May 30 '15 at 10:52
  • The first chapter of his book should already provide some insight. – Demosthene May 30 '15 at 11:12
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    @Omnomnomnom As far as I understand, the turning angle of a matrix A is the largest angle that A can rotate a vector. Even that is somewhat imprecise (e.g. a rotation about what?). It doesn't take a lot of google searches to notice that all paths seem to take you back to Karl Gustafson. – JessicaK May 30 '15 at 11:24
  • I just scanned the first chapter and didn't really see a definition. Does he ever explicitly define them anywhere. I take from your last comment, no? – muaddib May 30 '15 at 12:51
  • nm, I see the definition of the "angle of an operator in the original paper": https://projecteuclid.org/download/pdf_1/euclid.bams/1183529619 – muaddib May 30 '15 at 12:53
  • One additional issue I have is that the wikipedia page and page 9-10 of the linked book points to a huge swathe of applications. I can't tell if there is anything which truly useful or at least mathematically interesting. It all reads like a lot of abstract nonsense. – JessicaK May 30 '15 at 13:10
  • He's actually very well published so I don't know what to make of it: http://euclid.colorado.edu/~gustafs/ – muaddib May 30 '15 at 16:13
  • Love the question :) – mick Jan 20 '17 at 01:21
  • I believe the angle between $x$ and $y$ is typically defined by $\cos^{-1} \frac {x^Ty} {||x|| \cdot ||y||}$. Substituting $Ax$ for $y$ might give you the turning angle. – David Schneider-Joseph Mar 18 '17 at 22:10

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I'm very interested in this.

My intuitive take is that normal eigenvalues are those vectors associated with a linear transformation that undergo no rotation or sheer, and are scalar multiples of direction of the transformation.

If I imagine a field of randomly pointing vectors (like flags without wind) and then apply a transformation, the anti-eigenvectors are those vectors that undergo MAXIMUM rotation under the transformation.

It's actually more interesting in some ways than eigenvectors which get all the airtime in linear algebra. Because one might also (practically and intuitively) want to know the set of vectors that undergo maximum deformation or rotation under a transformation. That's where a lot of the "work" of the transform is happening.

Sean Kennedy
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