Activity4.2.4

  1. Identify the eigenvalues, and their multiplicities, of an \(n\times n\) matrix whose characteristic polynomial is \((2-\lambda)^3(-3-\lambda)^{10}(5-\lambda)\text{.}\) What can you conclude about the dimensions of the eigenspaces? What is the dimension of the matrix? Do you have enough information to guarantee that there is a basis of \(\real^n\) consisting of eigenvectors?

  2. Find the eigenvalues of \(\left[\begin{array}{rr} 0 \amp -1 \\ 4 \amp -4 \\ \end{array}\right]\) and state their multiplicities. Can you find a basis of \(\real^2\) consisting of eigenvectors of this matrix?

  3. Consider the matrix \(A = \left[\begin{array}{rrr} -1 \amp 0 \amp 2 \\ -2 \amp -2 \amp -4 \\ 0 \amp 0 \amp -2 \\ \end{array}\right]\) whose characteristic equation is

    \begin{equation*} (-2-\lambda)^2(-1-\lambda) = 0 \text{.} \end{equation*}
    1. Identify the eigenvalues and their multiplicities.

    2. For each eigenvalue \(\lambda\text{,}\) find a basis of the eigenspace \(E_\lambda\) and state its dimension.

    3. Is there a basis of \(\real^3\) consisting of eigenvectors of \(A\text{?}\)

  4. Now consider the matrix \(A = \left[\begin{array}{rrr} -5 \amp -2 \amp -6 \\ -2 \amp -2 \amp -4 \\ 2 \amp 1 \amp 2 \\ \end{array}\right]\) whose characteristic equation is also

    \begin{equation*} (-2-\lambda)^2(-1-\lambda) = 0 \text{.} \end{equation*}
    1. Identify the eigenvalues and their multiplicities.

    2. For each eigenvalue \(\lambda\text{,}\) find a basis of the eigenspace \(E_\lambda\) and state its dimension.

    3. Is there a basis of \(\real^3\) consisting of eigenvectors of \(A\text{?}\)

  5. Consider the matrix \(A = \left[\begin{array}{rrr} -5 \amp -2 \amp -6 \\ 4 \amp 1 \amp 8 \\ 2 \amp 1 \amp 2 \\ \end{array}\right]\) whose characteristic equation is

    \begin{equation*} (-2-\lambda)(1-\lambda)(-1-\lambda) = 0 \text{.} \end{equation*}
    1. Identify the eigenvalues and their multiplicities.

    2. For each eigenvalue \(\lambda\text{,}\) find a basis of the eigenspace \(E_\lambda\) and state its dimension.

    3. Is there a basis of \(\real^3\) consisting of eigenvectors of \(A\text{?}\)

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