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

[hellomrrobot]

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?

 

[Dr. Courtney]

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.

 

[hellomrrobot]

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]

https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

 

[HallsofIvy]

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.