# Interpretation of eigenvalues and associated eigenvalues

I am given a 4x4 matrix such and a basis for $$R^4$$, $$\{u_1,u_2,u_3,u_4\}$$. Suppose the following conditions: $$Bu_1 = 2u_1 \\Bu_2 = 0\\Bu_3 = u_4\\Bu_4 = u_3$$

First, I am asked to find the eigenvalues. It's fairly obvious that $$1$$ and $$2$$ are eigenvalues. Then I manipulate the lower 2 equations as such:

$$Bu_3 = u_4\\B(Bu_3) = Bu_4\\B^2u_3 = u_3$$

Hence an eigenvalue of $$B^2$$ is supposed to be 1. Square rooting this, I obtain eigenvalues of $$1$$ and $$-1$$ for $$B$$.

Thereafter, I need to find the associated eigenvectors. Again, for $$\lambda = 1$$ is $$u_1$$ and $$\lambda = 0$$ is $$u_2$$. But for the other two, I cannot figure it out. The answer key says that that of $$\lambda = 1$$ is $$u_3+u_4$$, and that of $$\lambda = -1$$ is $$u_3-u_4$$. I cannot comprehend why. Am I missing out on come manipulation?

Any advice is appreciated! Thank you!

$$Bu_3=u_4$$ and $$Bu_4=u_3$$ $$\Rightarrow$$ $$Bu_3+Bu_4=B(u_3+u_4)=u_4+u_3$$ hence 1 is an eigenvalue for $$u_3+u_4$$
$$Bu_3=u_4$$ $$\Rightarrow$$ $$-Bu_3=B(-u_3)=-u_4$$
$$B(-u_3)=-u_4$$ and $$Bu_4=u_3$$ $$\Rightarrow$$ $$B(u_3-u_4)=-(u_3-u_4)$$ hence -1 is an eigenvalue