How to say if Eigenvectors of A are orthogonal or not? without computing eigenvectors

Question

I am give matrix : $$A=\begin{bmatrix} 0&-1 & 2 \\ -1 & -1 & 1 \\ 2 & 1 &0 \end{bmatrix} $$

1. Without finding the eigenvalues and eigenvectors, determine whether the eigenvectors are orthogonal or not. Justify your answer

2. Express matrix $A$ in the form $A=UDU^T$ where $D$ is a diagonal matrix and $U$ is an orthogonal matrix. What are $U$ and $D$ ?

I can check if a vectors are orthogonal or not, by dot product = 0

I know that if $B^T=B^{-1}$ so that $B$ be can be said orthogonal, and $B^TB=I$

I also can find the eigenvalues and eigenvectors, but the question asks without finding them..

How to check whether eigenvalues are orthogonal or not without finding? and how to express $A=UDU^T$?

Answer 1

Let $v$ be an eigenvector correspond to $\lambda$ and let $w$ be an eigenvector correspond to $\delta$. Then $$\lambda \langle v,w \rangle= \langle \lambda v,w \rangle=\langle Av,w \rangle=\langle v,A^tw \rangle=\langle v,\delta w \rangle= \delta \langle v,w \rangle\Rightarrow (\lambda-\delta)\langle v,w \rangle \Rightarrow \langle v,w \rangle=0$$