Difference between revisions of "Eigenvalue"

Latest revision as of 21:11, June 26, 2010

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-{{merge|eigenvector}}
+#REDIRECT[[Eigenvectors and Eigenvalues]]
-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>
-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]].
-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>.
-[[category:Linear algebra]]