What Does the Book Say About the Eigenvalues of 3x3 Matrices?

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Discussion Overview

The discussion revolves around the eigenvalues of 3x3 matrices, specifically addressing the claims made in a book regarding the existence of eigenvalues for such matrices. Participants explore the implications of these claims in the context of real and complex matrices.

Discussion Character

  • Debate/contested
  • Technical explanation

Main Points Raised

  • Some participants note that the book states every 3x3 matrix has at least one eigenvalue, questioning whether this should specify at least one real eigenvalue.
  • One participant suggests that the book likely assumes real matrices, referencing the characteristic polynomial and its factorization in an algebraically closed field.
  • Another participant mentions that eigenvalues must belong to the base field and that a cubic polynomial with real coefficients always has a real root.
  • There is a query about the possibility of a complex 3x3 matrix having only one eigenvalue.
  • One participant provides an example of a diagonal matrix with repeated eigenvalues to illustrate the point.
  • Another participant requests more information about the book, suggesting that revealing the title or quoting it could clarify the discussion.

Areas of Agreement / Disagreement

Participants express differing views on whether the statement about eigenvalues should specify real eigenvalues, and there is uncertainty regarding the implications for complex matrices. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants highlight the need for clarity regarding the assumptions made in the book, particularly concerning the types of matrices (real vs. complex) and the nature of eigenvalues (distinct vs. repeated).

LagrangeEuler
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I found in one book that every quadratic matrix 3x3 has at least one eigenvalue. I do not understand. Shouldn't be stated at least one real eigenvalue? Thanks for the answer.
 
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LagrangeEuler said:
I found in one book that every quadratic matrix 3x3 has at least one eigenvalue. I do not understand. Shouldn't be stated at least one real eigenvalue? Thanks for the answer.
Yes. I assume that the book is primarily assuming real matrices.

We get a characteristic polynomial which decomposes into linear factors in case of an algebraic closed field. So we have ##\chi(t)=-(t-\lambda_1)(t-\lambda_2)(t-\lambda_3)##. But we do not have any knowledge whether the algebraic multiplicities are all one. E.g. ##\lambda_1=\lambda_2=\lambda_3## cannot be ruled out, what commonly is called one eigenvalue, not three.
 
Last edited:
By definition eigenvalues have to belong to the base field. A qubic polynomial with real quoeficients always has a real root.
 
Yes. But if we have complex 3x3 matrix is it possible to have only one eigenvalue?
 
LagrangeEuler said:
Yes. But if we have complex 3x3 matrix is it possible to have only one eigenvalue?
Like the diagonal matrix ##diag(i, i, i)##?
 
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LagrangeEuler said:
Yes. But if we have complex 3x3 matrix is it possible to have only one eigenvalue?
And what is the book talking about? May be if you revieled the title and the page or quoted the book, we wouldn't have to guess.
 

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