Difference between revisions of "Eigenvalue"
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:<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 eigenvalues are the zeroes of a matrix's [[characteristic polynomial]]. |
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| + | The product of all the eigenvalues of a matrix equals the value of the matrix's determinant. Since a matrix is invertible if and only if the determinant is non-zero, it is invertible if and only if zero is not an eigenvalue. | ||
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 14:58, May 2, 2010
An eigenvalue of a square matrix is a real number or complex number 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 eigenvalues are the zeroes of a matrix's characteristic polynomial.
The product of all the eigenvalues of a matrix equals the value of the matrix's determinant. Since a matrix is invertible if and only if the determinant is non-zero, it is invertible if and only if zero is not an eigenvalue.
The span of all the eigenvectors corresponding to a fixed eigenvalue is called the eigenspace of .
