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 , 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 , , where 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_i } is the corresponding eigenvalue of . For example:
Let ,
Then,
Thus, is an eigenvector of matrix , and its corresponding eigenvalue .
Resources
- Eigenvalues and Eigenvectors, MIT Math Department
- Eigenvalues and Eigenvectors, Wikipedia