Eigenvalue: Difference between revisions
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In [[linear algebra]], an '''eigenvector''' of a [[linear map]] <math>L</math> is a non-zero [[vector]] <math>\bold{v}</math> such that applying <math>L</math> to <math>\bold{v}</math> results in a vector in the same direction as <math>v</math> (including possibly the zero vector). In other words, <math>\bold{v}</math> is an eigenvector for <math>L</math> if and only if there is some scalar constant <math>\lambda</math> such that <math>L \bold{v} = \lambda \bold{v}</math>. Here, <math>\lambda</math> is known as the '''eigenvalue''' associated to the eigenvector. The '''eigenspace''' of an eigenvalue refers to the set of all eigenvectors that correspond with that eigenvalue, and is a [[vector space]]; in particular, it is a subspace of the domain of the map <math>L</math>. | In [[linear algebra]], an '''eigenvector''' of a [[linear map]] <math>L</math> is a non-zero [[vector]] <math>\bold{v}</math> such that applying <math>L</math> to <math>\bold{v}</math> results in a vector in the same direction as <math>v</math> (including possibly the zero vector). In other words, <math>\bold{v}</math> is an eigenvector for <math>L</math> if and only if there is some scalar constant <math>\lambda</math> such that <math>L \bold{v} = \lambda \bold{v}</math>. Here, <math>\lambda</math> is known as the '''eigenvalue''' associated to the eigenvector. The '''eigenspace''' of an eigenvalue refers to the set of all eigenvectors that correspond with that eigenvalue, and is a [[vector space]]; in particular, it is a subspace of the domain of the map <math>L</math>. | ||
Eigenvalues have many many properties that it enjoy. For example, it is used for [[Diagonalizability|diagonalizing matrices]] (Note that that page shows how to calculate eigenvalues and eigenvectors). | |||
Additionally, as a extra entry of the Invertible Matrix Theorem, <math>0</math> is a eigenvalue of a matrix if and only if that matrix is NOT invertible. | |||
==See Also== | |||
* [[Linear Algebra]] | |||
* [[Diagonalizability]] | |||
* [[Characteristic Equation]] | |||
{{stub}} | {{stub}} | ||
[[Category:Linear algebra]] | [[Category:Linear algebra]] | ||
Latest revision as of 20:22, 28 May 2025
In linear algebra, an eigenvector of a linear map 
is a non-zero vector 
such that applying to results in a vector in the same direction as 
(including possibly the zero vector). In other words, is an eigenvector for if and only if there is some scalar constant 
such that 
. Here, is known as the eigenvalue associated to the eigenvector. The eigenspace of an eigenvalue refers to the set of all eigenvectors that correspond with that eigenvalue, and is a vector space; in particular, it is a subspace of the domain of the map .
Eigenvalues have many many properties that it enjoy. For example, it is used for diagonalizing matrices (Note that that page shows how to calculate eigenvalues and eigenvectors).
Additionally, as a extra entry of the Invertible Matrix Theorem, 
is a eigenvalue of a matrix if and only if that matrix is NOT invertible.
See Also
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