Difference between revisions of "Eigenvectors and Eigenvalues"

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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.
 
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.
  
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==Eigenvalues==
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==Characteristic Property of Eigenvalues and Eigenvectors==
  
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An '''eigenvalue''' of a square [[matrix]] <math>A</math> is a scalar <math>\lambda</math> such that
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An '''eigenvalue''' of a square n &times; 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>
 
:<math>A\boldsymbol{x}=\lambda\boldsymbol{x}</math>
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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.
 
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.
  
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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.
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The same definition is valid for n &times; n matrices over any [[field]] '''F''': Then <math>\lambda \in \mathbf{F}</math> and <math>\boldsymbol{x}\in\mathbf{F}^n</math>.
  
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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.
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The eigenvectors represent directions that are preserved by linear transformations of a [[vector space]].
  
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==Eigenvectors==
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All the eigenvectors of a particular eigenvalue [[span]] a vector space called the [[eigenspace]].
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'''Eigenvectors''' represent directions that are preserved by linear transformations of a [[vector space]].
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An '''eigenvector''' of an ''n x n'' square [[matrix]] <math>A</math> is a vector <math>\boldsymbol{x}\in\mathbb{C}^n</math> such that,
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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.
  
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:<math>A\boldsymbol{x}=\lambda\boldsymbol{x}</math>
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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.
  
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where <math>\lambda</math> is [[complex number]], known as a [[eigenvalue]]. The eigenvalue is the zero of a matrix's [[charateristic polynomial]].
 
  
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All the eigenvectors of a particular eigenvalue [[span]] a vector space called the [[eigenspace]].
 
  
  
  
 
[[category:Linear algebra]]
 
[[category:Linear algebra]]

Revision as of 14:45, June 29, 2010

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 Eigenvalues and Eigenvectors

An eigenvalue of a square n × n matrix with real entries is a scalar such that

for some non-zero vector 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.

The same definition is valid for n × n matrices over any field F: Then and .

The 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.

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 is called the eigenspace of . The dimension of this space is called the geometric multiplicity of the eigenvalue.

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