Difference between revisions of "Eigenvalue"

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

Redirect to:

@@ Line 1: / Line 1: @@
-{{merge|eigenvector}}
+#REDIRECT[[Eigenvectors and Eigenvalues]]
-An '''eigenvalue''' of a square [[matrix]] <math>A</math> is a scalar <math>\lambda</math> such that
-:<math>A\boldsymbol{x}=\lambda\boldsymbol{x}</math>
-for some non-zero vector <math>\boldsymbol{x}\in\mathbb{R}^n</math> known as a [[eigenvector]]. The eigenvalues are the zeroes of a matrix's [[characteristic polynomial]], the degree of the corresponding root is called the '''algebraic multiplicity''' of the eigenvalue.
-If the characteristic polynomial splits into linear factors, then he product of all the eigenvalues of a matrix counted with their algebraic multiplicities 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 dimension of this space is called the '''geometric multiplicity''' of the eigenvalue.
-[[category:Linear algebra]]