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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 $A=UDU^T$?

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      $\begingroup$ Do you know the Spectral theorem? $\endgroup$ – Huy Jan 9 '15 at 10:40
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      $\begingroup$ The matrix is symmetric. $\endgroup$ – Git Gud Jan 9 '15 at 10:42
    • $\begingroup$ If the eigenvalues are not distinct then some of the eigenvectors are not orthogonal. However, there is an orthogonal basis for the vector space. $\endgroup$ – Ofir Schnabel Jan 9 '15 at 10:44
    • $\begingroup$ Spectral theorem is like $D$ diagonal matrix is just the main diagonal elements are $\lambda$s ? @Huy If so how i can find the orthogonal matrix $U$? $\endgroup$ – Andy Jan 9 '15 at 10:46
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      $\begingroup$ @Andy Do you know how to diagonalize a matrix? Then do it. When you find a basis of eigenvectors, simply apply G.S. to it and you'll get your $U$. $\endgroup$ – Git Gud Jan 9 '15 at 10:53
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    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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