Eigenvalues and Eigenvectors

Definition

In linear algebra, an eigenvector of a matrix is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by $\lambda$, is the factor by which the eigenvector is scaled. That is, given some eigenvector $v_{i}$ of a square matrix $\mathbf {A}$, $\mathbf {A} v_{i}=\lambda _{i}v_{i}$, where $\lambda _{i}$ is the corresponding eigenvalue of $v_{i}$. For example:

Let $\mathbf {A} ={\begin{bmatrix}3&4&-2\\1&4&-1\\2&6&-1\end{bmatrix}}$, $v_{1}={\begin{bmatrix}1\\1\\2\end{bmatrix}}$

$\mathbf {A} v_{1}={\begin{bmatrix}3&4&-2\\1&4&-1\\2&6&-1\end{bmatrix}}{\begin{bmatrix}1\\1\\2\end{bmatrix}}={\begin{bmatrix}3\\3\\6\end{bmatrix}}=3{\begin{bmatrix}1\\1\\2\end{bmatrix}}=3v_{1}$

Thus, $v_{1}$ is an eigenvector of matrix $\mathbf {A}$, and its corresponding eigenvalue $\lambda _{1}=3$.

Resources

• Eigenvalues and Eigenvectors, MIT Math Department
• Eigenvalues and Eigenvectors, Wikipedia
• Solving for Eigenvalues of 2x2 Matrix, Khan Academy
• Eigenvalues of a 3x3 Matrix, Khan Academy
• Finding Eigenvalues and Eigenvectors: 2x2 Matrix Example, patrickJMT
• Eigenvalues and Eigenvectors of a 3x3 Matrix,