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 $A=UDU^T$?

  • $\endgroup$
    • 1
      $\begingroup$ Do you know the Spectral theorem? $\endgroup$ – Huy Jan 9 '15 at 10:40
    • 1
      $\begingroup$ 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
    • 1
      $\begingroup$ @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

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


    Your Answer

    By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

    Not the answer you're looking for? Browse other questions tagged or ask your own question.