What is the significance of Eigenvalues in a 2 by 2 matrix?

[rocomath]

Assuming 2 by 2.

Ok, I'm asked to find the Eigenvalues. How do I know which should be lambda 1 and lambda 2? I can find the lambda's easily, but does it matter which is 1 or 2? It becomes important when I'm asked to diagonalize.

$$A=S\Lambda S^{-1}$$

 

[matt grime]

The order is not important when you diagonalize.

 

[rocomath]

The order is not important when you diagonalize.

Thanks matt! Now I can go back to studying :)

 

[HallsofIvy]

Swapping $\lambda_1$ and $\lambda_2$ will swap their positions on the diagonal but either way is a valid diagonalization.