Diagonalization

 

  1. Use the factorization \(A=PDP^{-1}\) to compute \(A^{k}\), where \(k\) represents an arbitrary positive integer.  solution

    2. \(\left[\begin{array}{lll}2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2\end{array}\right]=\left[\begin{array}{rrr}1 & 1 & 2 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right]\left[\begin{array}{lll}5 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right] \left[\begin{array}{rrr}1 / 4 & 1 / 2 & 1 / 4 \\ 1 / 4 & 1 / 2 & -3 / 4 \\ 1 / 4 & -1 / 2 & 1 / 4\end{array}\right]\)

     

  2. Diagonalize the matrix, if possible:  \(A=\left[\begin{array}{rr}3 & -1 \\1 & 5 \end{array}\right]\).   solution

    3.  

  3. Determine if the matrix is diagonalizable:  \(A=\left[\begin{array}{rr}3 & 1 \\3 & 5 \end{array}\right]\).   solution

    4.  

  4. Diagonalize the matrix, if possible:   \(A=\left[\begin{array}{rrrr}5 & 0 & 0 \\ 0 & 5 & 0 \\ 1 & 4 & -3 \end{array}\right]\).  solution

    5.  

  5. Diagonalize the matrix, if possible:   \(A=\left[\begin{array}{rrr} 0 & -4 & -6 \\ -1 & 0 & -3 \\ 1 & 2 & 5 \end{array}\right]\),   given that the eigenvalues of \(A\) are \(1, 2\).   solution


some short-answer questions  solution