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I was wondering how the QR algorithm could be used to calculate the eigenvalues of matrices containing elements which are members of fields other than R and C. For example, say we have a matrix of functions, these functions all elements of a field (and either never or always zero, maybe exponentials?), or a matrix with elements being members of some other field. If we applied the QR algorithm to this matrix (I would guess this is possible, as Householder reflectors need only multiplication to work, for example, and if we can divide, as in a field, we can generate appropriate reflectors), would it converge?

What conditions, if any, must be imposed on a field other than the field conditions themselves to guarantee/make likely convergence? Is an algebra of some sort necessary (for roots of the characteristic polynomial to exist)?

My background is primarily numerical, and even then not very extensive, so please answer with this in mind!

EDIT: I've removed my reference to finite fields, as this has been answered in the comments below. That said, I am still curious as to what properties a field must have to make the QR algorithm converge.

tekne
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  • Welcome to MSE. Please use MathJax. – José Carlos Santos Jul 29 '17 at 17:00
  • A subtlety you may have missed is that a Householder reflection needs the concept of a unit vector. This does not exist over a finite field (or many other fields - in particular those of positive characteristic). The usual inner product of a non-zero vector with itself may be zero. Or a non-square non-zero when dividing it by the square root of the inner product may not be possible. The same problem kills all hope of Gram-Schmidt orthogonalization. – Jyrki Lahtonen Jul 30 '17 at 06:03
  • Also, finite fields don't have a useful topology, so it is unclear to me what you mean by "convergence". – Jyrki Lahtonen Jul 30 '17 at 06:04
  • @Jyrki that's very interesting, and in part answers my question. That said, I still want to know what qualities are necessary for the QR algorithm to work for a field. You make an interesting point about topology, I didn't consider that (I've only ever worked with R and C, trying to expand my knowledge). I will edit my question accordingly. Thanks! – tekne Jul 30 '17 at 17:53
  • See https://math.stackexchange.com/questions/4847409 for an example ($\mathbb F_2$) where matrices need not possess a QR decomposition. – joriki Jan 19 '24 at 15:32

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