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Eigenvectors and Eigenvalues (view source)
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{{Jargon}}
{{Jargon}}
==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.
==Eigenvectors==
'''Eigenvectors''' represent directions that are preserved by linear transformations of a [[vector space]].
'''Eigenvectors''' represent directions that are preserved by linear transformations of a [[vector space]].
All the eigenvectors of a particular eigenvalue [[span]] a vector space called the [[eigenspace]].
All the eigenvectors of a particular eigenvalue [[span]] a vector space called the [[eigenspace]].
[[category:Linear algebra]]
[[category:Linear algebra]]