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Wize University Linear Algebra Textbook > Eigenvalues and Eigenvectors (Spectral Theory)

Eigenvalues and Eigenvectors

4 Activities

The matrix
  ⁣A‾=[5−6−6−1423−6−4]\ul{A} = \begin{bmatrix} 5 & -6 & - 6 \\ -1 & 4 & 2 \\ 3 & -6 & -4 \end{bmatrix}

has characteristic polynomial
CA(x)=x3−5x2+8x−4=(x−1)(x−2)2C_{A}(x) = x^3 - 5x^2 + 8x - 4 = (x-1)(x-2)^2.

  1. Give the eigenvalues of A‾\A, then
  2. determine their algebraic and geometric multiplicity;
  3. finally,
  4. determine whether or not A‾\A is diagonalizable, and
  5. if it is, give the matrices P ⁣‾ \matP and D ⁣‾ \D, such that   ⁣A‾=P ⁣‾  D ⁣‾  P−1\ul{A} = \matP \, \D \, \minv{P} .

If λ1≤λ2≤λ3\lambda_1 \leq \lambda_2 \leq \lambda_3 (i.e. if we order the eigenvalues from the lowest value to highest value), then:
More Eigenvalues and Eigenvectors Questions:
Eigenvalues and Eigenvectors
Practice: Eigenvalues and Eigenvectors
Find all eigenvalues and eigenvectors of the matrix A=[010010−111]A=\left[\begin{array}{rrr} 0&1&0\\ 0&1&0\\ -1&1&1 \end{array}\right].
Eigenvalues and Eigenvectors
Practice: Eigenvalues and Eigenvectors
Determine which of the following vectors are eigenvectors of A=[133−3−5−3331]A=\left[\begin{array}{rrr} 1&3&3\\-3&-5&-3\\3&3&1 \end{array}\right].
[Select all that apply]
133 - FML 2 - 18.1W e.g. 6
Find the characteristic polynomial CA‾(x)C_{\A}(x) for the matrix   ⁣A‾=[31−122−1220]\ul{A} = % \begin{bmatrix} 3 & 1 & -1 \\ 2 & 2 & -1 \\ 2 & 2 & 0 \\ \end{bmatrix}.
Practice Problems
The matrix
A‾=[31−122−1220]\A=\begin{bmatrix} 3&1&-1\\ 2&2&-1\\ 2&2&0 \end{bmatrix}
has characteristic polynomial:
19.4F_Final_Builder_Ch_12.7_Dynamical_Systems_$\tkcth{eg6}$_$\key{Final}$_Builder_$\tkcth{12.7.}\tkcf{6}$_
ξ⃗1=[31]\vXi_1 = \colvec{3}{1} and ξ⃗2=[12]\vXi_2 = \colvec{1}{2} are eigenvectors for the matrix
A‾=[53−212] \A = \sm{5}{3}{-2}{12}
where λ1=6\lambda_1 = 6 and λ2=11\lambda_2 = 11.
19.4F_Final_Builder_Ch_12.7_Dynamical_Systems_$\tkcth{eg5}$_$\key{Final}$_Builder_$\tkcth{12.7.}\tkcf{5}$_
Find the eigenvectors and corresponding eigenvalues of the matrix
A‾=[53−212] \A = \sm{5}{3}{-2}{12}
19.4F_Final_Builder_Ch_12.7_Dynamical_Systems_$\tkcth{eg9}$_$\key{Final}$_Builder_$\tkcth{12.7.}\tkcf{9}$_
Show that if x⃗\vx is an eigenvector of   ⁣M ⁣‾  \M with eigenvalue kk, then it is also an eigenvector of the matrix B ⁣‾ =10  ⁣M ⁣‾  \B = 10\M.
Use your result to find the eigenvalue of x⃗\vx with respect to the matrix B ⁣‾ \B, i.e. B ⁣‾  x⃗=λ x⃗\B \, \vx = \lambda \, \vx where λ=\lambda =
10k
19.4F_Final_Builder_Ch_12.6_Extra_Eigen_Things_$\tkcth{eg8}$_$\key{Final}$_Builder_$\tkcth{12.6.}\tkcf{8}$_
Find a basis for the eigenspace of the matrix A‾=[00−2121103]\A = \begin{bmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3 \end{bmatrix} corresponding to the eigenvalue λ2=2\lambda_2=2 (i.e. find the vector(s) that span the space A‾x⃗=2x⃗\A \vx = 2\vx).
19.4F_Final_Builder_Ch_12.6_Extra_Eigen_Things_$\tkcth{eg7}$_$\key{Final}$_Builder_$\tkcth{12.6.}\tkcf{7}$_
Find a basis for the eigenspace of the matrix
A‾=[00−2121103]\A = \begin{bmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3 \end{bmatrix}
corresponding to the eigenvalue λ1=1\lambda_1=1 (i.e. find the vector(s) that span the space A‾x⃗=2x⃗\A \vx = 2\vx).
19.4F_Final_Builder_Ch_12.6_Extra_Eigen_Things_$\tkcth{eg6}$_$\key{Final}$_Builder_$\tkcth{12.6.}\tkcf{6}$_
Find the eigenvalues of the matrix A‾=[00−2121103]\A = \begin{bmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3 \end{bmatrix}.
19.4F_Final_Builder_Ch_12.6_Extra_Eigen_Things_$\tkcth{eg4}$_$\key{Final}$_Builder_$\tkcth{12.6.}\tkcf{4}$_
Find the eigenvalues and corresponding eigenvectors of the matrix A‾=[−1320].\A = \sm{-1}{3}{2}{0}.
19.4F_Final_Builder_Ch_12.6_Extra_Eigen_Things_$\tkcth{eg3}$_$\key{Final}$_Builder_$\tkcth{12.6.}\tkcf{3}$_
Find the eigenvector of the matrix A‾=[308−1]\A = \sm{3}{0}{8}{-1} corresponding to the eigenvalue λ2=3\lambda_2 = 3.
19.4F_Final_Builder_Ch_12.6_Extra_Eigen_Things_$\tkcth{eg1}$_$\key{Final}$_Builder_$\tkcth{12.6.}\tkcf{1}$_
Find the eigenvalues of the matrix A‾=[308−1].\A = \sm{3}{0}{8}{-1}.
Practice: Eigenvalues and Eigenvectors
Given A=[6−31211111]A=\begin{bmatrix}6&-3&1\\2&1&1\\1&1&1\end{bmatrix} and the eigenvector [875]T\begin{bmatrix}8&7&5\end{bmatrix}^T, find the eigenvalue to which it is associated.
$\tkct{Mock F1}$ $\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W - e.g. 27
What are the eigenvectors of the standard matrix representation of the transformation T\bcb{T} that projects a vector v⃗\bcb{\vec{v}} onto the line x+y=0\bcb{x + y = 0}?
NB it does not matter which vector / value is called "1" or "2", so long as the vector and value you choose, match. If there is no way to create a match, select the appropriate option.
Vector Space and Subspace