Let C= [a b; b a]. Show that [1; 1] and [1; -1] are eigenvectors of C. What are corresponding eigenvalues?

Question

Let C= [a b; b a]. Show that [1; 1] and [1; -1] are eigenvectors of C. What are corresponding eigenvalues?

I am unsure on how to do this problem.

Answer 1

So we have to prove that

$$\begin{bmatrix}a&b\\b&a\end{bmatrix}\begin{bmatrix}\ \ \ 1\\-1\end{bmatrix}=\lambda_1\begin{bmatrix}\ \ \ 1\\-1\end{bmatrix}\tag 1$$ and that $$\begin{bmatrix}a&b\\b&a\end{bmatrix}\begin{bmatrix}\ 1\\1\end{bmatrix}=\lambda_2\begin{bmatrix}1\\1\end{bmatrix}\tag 2$$ where $\lambda_1$ and $\lambda_2$ are the eigenvalues of $\left[\begin{smallmatrix}a&b\\b&a\end{smallmatrix}\right]$.

So, calculate the eigenvalues. The characteristic equation is

$$(a-\lambda)^2-b^2=0$$

and its solutions are

$$\lambda_{1,2}=a\pm b.$$

You can check that $(1)$ and $(2)$ hold.