Eigenvalues and Eigenvectors   Lecture 1   Lecture 2

 

  1. Is \( \left[\begin{array}{r}1 \\ -2 \\ 1\end{array}\right]\) an eigenvector of \(\left[\begin{array}{lll}3 & 6 & 7 \\ 3 & 3 & 7 \\ 5 & 6 & 5\end{array}\right]\) ? If so, find the eigenvalue.   solution

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  2. Find a basis for the eigenspace corresponding to the given eigenvalue. \( \quad A=\left[\begin{array}{cc}10 & -9 \\ 4 & -2\end{array}\right], \: \lambda=4. \quad \) solution

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  3. Find a basis for the eigenspace corresponding to the given eigenvalue. \( \quad A=\left[\begin{array}{rrr} 4 & 0 & 1 \\ -2 & 1 & 0 \\ -2 & 0 & 1 \end{array}\right], \: \lambda=1. \quad \) solution

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  4. Find the eigenvalues and eigenvectors of \( \quad A=\left[\begin{array}{rr} 2 & 3 \\ 3 & -6\end{array}\right] \quad \) solution

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  5. Find the eigenvalues and eigenvectors of \( \quad A=\left[\begin{array}{cc}5 & 3 \\ 4 & 4\end{array}\right] \quad \) solution

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  6. Without calculation find eigenvalue(s) of the following matrices.   solution

    7. \(A=\left[\begin{array}{rrr}4 & 0 & 0 \\ -2 & 1 & 0 \\ 3 & 1 & 3\end{array}\right], \quad B=\left[\begin{array}{rrr}3 & 2 & 1 \\ 3 & 2 & 1 \\ 2 & 1 & -5\end{array}\right] \)

     

  7. Show that if \(A=QR\) with \(Q\) invertible, then \(A\) is similar to \(A_1=RQ\).  Proof


  8. Show that if A and B are similar matrices, then det(A) = det(B).  Proof


  9. Find the eigenvalues of \(A = \left[\begin{array}{rrr}5 & -2 & 3 \\ 0 & 2 & 0 \\ 6 & 7 & -2\end{array}\right].\quad\) solution

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  10. Find the eigenvalues and eigenvectors of \(A = \left[\begin{array}{rrr} 1 & 0 & 3 \\ -3 & 4 & 1 \\ 0 & 0 & 2\end{array}\right].\quad\) solution