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

diagonalizationlinear algebramatricesmatrix decomposition

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 $A=UDU^T$?

    • Best Answer

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

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