What is the relationship between eigenvalues and eigenvectors in 3x3 matrices?

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Discussion Overview

The discussion revolves around the relationship between eigenvalues and eigenvectors in 3x3 matrices, specifically addressing the claim that the eigenvalues of one matrix are the square roots of the eigenvalues of another matrix while sharing the same eigenvectors. The scope includes conceptual clarification and mathematical reasoning.

Discussion Character

  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that if two 3x3 matrices have the same eigenvectors, then the eigenvalues of one are the square roots of the eigenvalues of the other.
  • One participant questions what this relationship looks like in practice, indicating a need for further explanation or examples.
  • Another participant provides a definition of eigenvalues and eigenvectors, illustrating with specific examples of matrices A and B, but does not clarify the relationship further.

Areas of Agreement / Disagreement

Participants generally agree on the basic premise that the eigenvectors are the same and that the eigenvalues relate as stated, but the discussion does not resolve the implications or visual representation of this relationship.

Contextual Notes

Some participants express uncertainty about the practical implications of the relationship between eigenvalues and eigenvectors, indicating a potential gap in understanding or application.

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What does it mean when it says eigenvalues of Matrix (3x3) A are the square roots of the eigenvalues of Matrix (3x3) B and the eigenvectors are the same for A and B?
 
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Seems simple to me.

You have two 3x3 matrices. Their eigenvectors are the same. The eigenvalues of one are the square roots of the eigenvalues of the other.
 
Dr. Courtney said:
Seems simple to me.

You have two 3x3 matrices. Their eigenvectors are the same. The eigenvalues of one are the square roots of the eigenvalues of the other.
Yes, but what does that look like? It has been a while since I have even used the word eigenvalue/vector...
 
Dr. Courtney said:
You have two 3x3 matrices. Their eigenvectors are the same. The eigenvalues of one are the square roots of the eigenvalues of the other.
So your question is really "what are eigenvalues and eigenvectors?". An "eigenvector for matrix A, corresponding to eigenvalue \lamba, is a vector, v, such that Av= \lambda v".

Suppose A= \begin{bmatrix}3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 6\end{bmatrix}. Then it is easy to see that 3 is an eigenvalue of A with eigenvector any multiple of (1, 0, 0), 2 is an eigenvalue of A with eigenvector any multiple of (0, 1, 0), and 6 is an eigenvalue with eigenvector any multiple of (0, 0, 1).

Similarly, let B= \begin{bmatrix}9 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 36\end{bmatrix}. Now, 9, 4, and 36 are eigenvalues of B with the same eigenvectors.
 

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