Difference between revisions of "Eigenvalues and Eigenvectors"
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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 <math> \lambda </math>, is the factor by which the eigenvector is scaled. That is, given some eigenvector <math> v_i </math> of a square matrix <math> | 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 <math> \lambda </math>, is the factor by which the eigenvector is scaled. That is, given some eigenvector <math> v_i </math> of a square matrix <math> | ||
\mathbf{A}</math>, <math> \mathbf{A}v_i = \lambda_i v_i</math>, where <math> \lambda_i </math> is the corresponding eigenvalue of <math> v_i </math>. For example: | \mathbf{A}</math>, <math> \mathbf{A}v_i = \lambda_i v_i</math>, where <math> \lambda_i </math> is the corresponding eigenvalue of <math> v_i </math>. For example: | ||
Revision as of 14:26, 24 September 2021
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 , is the factor by which the eigenvector is scaled. That is, given some eigenvector of a square matrix , , where is the corresponding eigenvalue of . For example:
Let , 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_1 = \begin{bmatrix} 1\\ 1\\ 2 \end{bmatrix}}
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 \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 (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_1 } is an eigenvector of matrix 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 \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
