Eigenvalues and Eigenvectors

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda } , is the factor by which the eigenvector is scaled. That is, given some eigenvector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_1 = 3 } .

Resources

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