Relationship between eigenvalues of A symmetric matrices

Question

Let $$A=\begin{pmatrix}a & b\\b & c\end{pmatrix} \in M_2\mathbb{(R)}$$

i) Find the eigenvalues of $A$

ii) If $\begin{pmatrix}1\\2\end{pmatrix}$ is an eigenvector of $A$, prove that $\begin{pmatrix}-2\\1\end{pmatrix}$ is an eigenvector of A.

For the first part, by using quadratic formula I get, $$\lambda=\frac{a+c\pm\sqrt{(a-c)^2+4b^2}}{2}$$

Please help me check whether the eigenvalues I get is correct or not. If yes, how can I relate this eigenvalue with the eigenvectors given?

Answer 1

If $\begin{pmatrix}1\\2\end{pmatrix}$ is an eigenvector, then:

$$\begin{pmatrix}a+2b\\b+2c\end{pmatrix}=\lambda\begin{pmatrix}1\\2\end{pmatrix}$$

So $\lambda = a+2b$ and $b+2c=2\lambda = 2(a+2b)$ or $2a+3b-2c=0$.

Now $$A\begin{pmatrix}-2\\1\end{pmatrix} = \begin{pmatrix}-2a+b\\-2b+c\end{pmatrix}$$

And you need to show that $-2a+b = -2(-2b+c)$.

Answer 2

for part II, use the fact that symmetric matrices have a full set of orthogonal eigenvectors spanning the whole space $R^n.$ this theorem is called the spectral decomposition theorem for symmetric matrices. anyway, if you have one eigenvector $(1,2)^T$ then $(-2,1)^T$ is orthogonal to the first one, therefore must be another eigenvector of the matrix.

edit: thomas andrews has a better answer that does not resort to the spectral theorem in the case of $2 \times 2$ symmetric matrix.