In [[linear algebra]], when a transformation of a space is carried out, some vectors (points in the space) are not rotated, but only extended or shrunk (moved farther or closer to the origin). The vectors are called the '''eigenvectors''' of the transformation, and the amount of extension or shrinkage carried out on that eigenvector is called the '''eigenvalue''' of the transformation corresponding to the given eigenvector.
==Characteristic Property of Eigenvaluesand Eigenvectors==
An '''eigenvalue''' of a square n × n [[matrix]] with real entries <math>A</math> is a scalar <math>\lambda\in \mathbb{R}</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 The same definition is invertible if valid for n × n matrices over any [[field]] '''F''': Then <math>\lambda \in \mathbf{F}</math> and only if the determinant is non-zero, it is invertible if and only if zero is not an eigenvalue<math>\boldsymbol{x}\in\mathbf{F}^n</math>.
The [[span]] eigenvectors represent directions that are preserved by linear transformations of all the eigenvectors corresponding to a fixed eigenvalue <math>\lambda</math> is called the [[eigenspacevector space]] <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 All the eigenvectors of a particular eigenvalue [[span]] a vector spacecalled the [[eigenspace]].
An '''eigenvector''' If the characteristic polynomial splits into linear factors, then he product of an 'all the eigenvalues of a matrix counted with their algebraic multiplicities equals the value of the matrix'n x n'' square [[s determinant. Since a matrix]] <math>A</math> is a vector <math>\boldsymbol{x}\in\mathbb{C}^n</math> such thatinvertible 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>A\boldsymbol{x}=\lambda</math> is called the [[eigenspace]] <math>E_\boldsymbol{x}lambda</math> of <math>A</math>. The dimension of this space is called the '''geometric multiplicity''' of the eigenvalue.
where <math>\lambda</math> is [[complex number]], known as a [[eigenvalue]]. The eigenvalue is the zero of a matrix's [[charateristic polynomial]].
All the eigenvectors of a particular eigenvalue [[span]] a vector space called the [[eigenspace]].
[[category:Linear algebra]]
