Finding eigenvalues and eigenspaces with only this info

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Discussion Overview

The discussion revolves around the problem of finding the remaining eigenvalues and eigenspaces of a 3x3 matrix 'A' given only one known eigenvalue and one known eigenvector that do not correspond to each other. The scope includes theoretical considerations and mathematical reasoning related to eigenvalues and eigenvectors.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions the feasibility of finding other eigenvalues and eigenspaces with only one eigenvalue and one eigenvector, noting the disparity in the amount of information available versus needed.
  • Another participant emphasizes that even if the known eigenvalue and eigenvector correspond, there could be infinitely many matrices that fit the criteria, suggesting that the known information does not constrain the other eigenvalues and eigenvectors.
  • A participant clarifies that the matrix 'A' is unknown and expresses a desire to find eigenvalues and eigenspaces without needing the eigenvectors.
  • One participant argues that knowing the eigenspaces allows for the selection of a set of eigenvectors, implying that the previous statement about finding eigenvalues without eigenvectors lacks meaning.
  • Another participant reiterates that a single eigenvalue and eigenvector provide no information about the other eigenvalues and eigenvectors, regardless of whether they correspond.
  • A later reply explores the hypothetical scenario of choosing arbitrary eigenvalues and eigenvectors to accompany the known ones, seeking a method for doing so.
  • One participant suggests selecting the remaining eigenvalues and eigenvectors to span R^3 and then calculating the matrix using the standard diagonalization theorem.

Areas of Agreement / Disagreement

Participants generally disagree on the ability to determine other eigenvalues and eigenspaces from a single eigenvalue and eigenvector. There are competing views on the implications of the known information and its limitations.

Contextual Notes

The discussion highlights limitations in the information provided, specifically the lack of constraints on the other eigenvalues and eigenvectors given only one eigenvalue and eigenvector. The dependence on definitions and assumptions regarding eigenvalues and eigenspaces is also noted.

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Lets say I have a 3x3 matrix 'A' and one known eigenvalue 'z' and one known eigenvector 'x', but they don't "belong" to each other, as in Ax =/= zx
Is there a way of finding the other eigenvalues and eigenspaces of A using only this piece of information?
Thanks.
 
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I doubt it.

Just count the amount of information needed and available.
A 3x3 matrix has 9 numbers.
An eigenvalue is one number.
An eigenvector is two numbers (it's only defined upto a scale factor)
Thus in total you have 3<9 pieces of information.

Note that the full eigensystem is 3 eigenvalues and 3 eigenvectors
= 3*1 + 3*2 = 3*(1+2) = 9 pieces of information.
 
Even if the eigenvalue and eigenvector "belong" to each other, the other eigenvalues and eigenvectors could be anything. There will exist an infinite number of matrices having the given eigenvector and eigenvalue.

Or are you saying that you have a known matrix, one eigenvalue and eigenvector, and want to know if there is a simpler way of finding the other eigenvalues and eigenvectors than the usual "solve the characteristic equation"?
 
No, the matrix A is unknown as well. And I only want to find the eigenvalues and eigenspaces of A, not necessarily its eigenvectors.
 
If you know the eigenspaces then you can choose a set of eigenvectors. So, your last statement is a little meaningless.
 
As I said before, given one eigenvalue and one eigenvector, whether they correspond or not, there exist an infinite number of matrices with those given eigenvalue and eigenvector and whatever other eigenvalues and eigenvectors you want to assign.

A single eigenvector and eigenvalue tell you nothing about other eigenvalues and eigenvectors.
 
Ok, I see, thanks.
But say we could choose arbitrary eigenvalues and eigenvectors that would go with the ones we already know. How would one go about doing that?
 

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