Eigenvalues and Eigenvectors

The Essentials

Eigenvalues and Eigenvectors are scalars and vectors that have this property:

\[ A\vec{x} =λ\vec{x} \]

Only square matrices have eigenvalues and eigenvectors. An n by n matrix will have n eigenvalues that aren’t necessarily unique. Each unique eigenvalue has one eigenvector associated with it, though the eigenvector isn’t unique and can be scaled. The number of linearly independent eigenvectors associated with repeated eigenvalues is between one and the number of repeated eigenvalues. For example, if a matrix has four eigenvalues \( λ_1 \), \( λ_2 \), \( λ_2 \), \( λ_2 \) then there will be one eigenvector associated with \( λ_1 \) and one, two, or three linearly independent eigenvectors associated with \( λ_2 \).

To find the eigenvalues of a matrix this equation is used:

\[ \det(A-\lambda I)=0 \]
To find the eigenvectors, this equation is used for each eigenvalue:
\[ [A-\lambda I]\vec{x}=0 \]

Example

Find the eigenvalues and eigenvectors for this matrix:

\[ \begin{bmatrix} -1 & 4/3 & 1\\ 0 & 3 & 3\\ -6 & 2 & -2 \end{bmatrix} \]

First we use the equation: \( \det(A-\lambda I)=0 \):

\[ \begin{bmatrix} -1-\lambda & 4/3 & 1\\ 0 & 3-\lambda & 3\\ -6 & 2 & -2-\lambda \end{bmatrix} \]
\[ \det(A-\lambda I) = (-1-\lambda) ((3-\lambda)(-2-\lambda)-6) - \frac{4}{3}(0+18)+1(0-(-6(3-\lambda))) = 0 \]
\[ -\lambda^3+13\lambda+12-24+18-6\lambda=0 \]
\[ -\lambda^3+7\lambda+6=0 \]
\[ \lambda=-1,3,-2 \]

Then, to find the eigenvector associated with \( \lambda_1=-1 \):

\[ \begin{bmatrix} - 1 - \lambda & 4 / 3 & 1\\ 0 & 3 - \lambda & 3\\ - 6 & 2 & - 2 - \lambda \end{bmatrix} \vec{x}=0 \]
\[ \begin{bmatrix} - 1 + 1 & 4 / 3 & 1\\ 0 & 3 + 1 & 3 \\ -6 & 2 & -2+1 \end{bmatrix} \vec{x}=0 \]
\[ \begin{bmatrix} 0 & 4/3 & 1\\ 0 & 4 & 3\\ -6 & 2 & -1 \end{bmatrix}\vec{x}=0 \]

Next we find the null space of this matrix which turns out to be:

\[ \vec{x}_1= \begin{bmatrix} 5\\ 9\\ -12 \end{bmatrix} \]

Using the same method with the other two eigenvalues gives us:

\[ \vec{x}_2= \begin{bmatrix} 1\\ 3\\ 0 \end{bmatrix}, \vec{x}_3= \begin{bmatrix} 5\\ 3\\ -5 \end{bmatrix} \]

Practice

Find the eigenvalues and eigenvectors for these matrices:

  1. \[ \begin{bmatrix} 2 & 6\\ 1 & 1 \end{bmatrix} \]
  2. \[ \begin{bmatrix} -4 & 5\\ 4 & 4 \end{bmatrix} \]

Solutions:

  1. \( \lambda_1=4, \vec{x}_1= \begin{bmatrix}3\\ 1\end{bmatrix}, \lambda_2=-1, \vec{x}_2= \begin{bmatrix}2\\ -1\end{bmatrix} \)
  2. \( \lambda_1=6, \vec{x}_1= \begin{bmatrix}1\\ 2\end{bmatrix}, \lambda_2=-6, \vec{x}_2= \begin{bmatrix}5\\ -2 \end{bmatrix} \)