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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbf {A} } , $\mathbf {A} v_{i}=\lambda _{i}v_{i}$, where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v_{1}} is an eigenvector of matrix Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbf {A} } , and its corresponding eigenvalue Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lambda _{1}=3} .

Resources

• Eigenvalues and Eigenvectors, MIT Math Department
• Eigenvalues and Eigenvectors, Wikipedia