Difference between revisions of "Eigenvalue"
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{{merge|eigenvector}} | {{merge|eigenvector}} | ||
| − | An '''eigenvalue''' of a square [[matrix]] <math>A</math> is a [[complex number]] <math>\lambda</math> such that | + | An '''eigenvalue''' of a square [[matrix]] <math>A</math> is a [[real|real number]] or [[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 eigenvalue is the zero of a matrix's [[ | + | 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 [[characteristic polynomial]]. |
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 17:18, May 1, 2010
It has been suggested that this article or section be merged with [[::eigenvector|eigenvector]]. (Discuss)
An eigenvalue of a square matrix <math>A</math> is a real number or complex number <math>\lambda</math> such that
- <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 eigenvalue is the zero of a matrix's characteristic polynomial.
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>.