Difference between revisions of "Eigenvalue"
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{{merge|eigenvector}} | {{merge|eigenvector}} | ||
| − | An '''eigenvalue''' of a square [[matrix]] <math>A</math> is a | + | An '''eigenvalue''' of a square [[matrix]] <math>A</math> is a [[complex number]] <math>\lambda</math> such that |
:<math>A\boldsymbol{x}=\lambda\boldsymbol{x}</math> | :<math>A\boldsymbol{x}=\lambda\boldsymbol{x}</math> | ||
| − | for some non-zero vector <math>\boldsymbol{x}\in\mathbb{R}</math> known as a [[eigenvector]]. The | + | for some non-zero vector <math>\boldsymbol{x}\in\mathbb{R}</math> known as a [[eigenvector]]. The eigenvalue is the zero of a matrix's [[charateristic polynomial]]. |
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The [[span]] of all the eigenvectors corresponding to a fixed eigenvalue <math>\lambda</math> is called the [[eigenspace]] <math>E_\lambda</math> of <math>A</math>. | The [[span]] of all the eigenvectors corresponding to a fixed eigenvalue <math>\lambda</math> is called the [[eigenspace]] <math>E_\lambda</math> of <math>A</math>. | ||
[[category:Linear algebra]] | [[category:Linear algebra]] | ||
Revision as of 16:46, May 2, 2010
An eigenvalue of a square matrix is a complex number 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} such that
for some non-zero vector 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 \boldsymbol{x}\in\mathbb{R}} known as a eigenvector. The eigenvalue is the zero of a matrix's charateristic polynomial.
The span of all the eigenvectors corresponding to a fixed 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} is called the eigenspace 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 E_\lambda} of 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 A} .