Eigenvector: Difference between revisions
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[[Category:Linear Algebra]] | [[Category:Linear Algebra]] | ||
Given a [[matrix]], its '''eigenvectors''' are special vectors that satisfy the following property: | |||
<math> | |||
A\vec{x}=\lambda\vec{x} | |||
</math> | |||
where <math>\lambda</math> is the '''eigenvalue''' associated with the eigenvector <math>\vec{x}</math> | |||
The definition of eigenvectors are also frequently written in this form: | |||
<math> | |||
(A-\lambda I)\vec{x}=0 | |||
</math> | |||
= Intuition = | |||
If we think of a matrix as a linear transformation, eigenvectors do not change direction. Instead, they simply scale by an eigenvalue. | |||
Revision as of 06:35, 10 June 2024
Given a matrix, its eigenvectors are special vectors that satisfy the following property:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\vec{x}=\lambda\vec{x} }
where is the eigenvalue associated with the eigenvector
The definition of eigenvectors are also frequently written in this form:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A-\lambda I)\vec{x}=0 }
Intuition
If we think of a matrix as a linear transformation, eigenvectors do not change direction. Instead, they simply scale by an eigenvalue.
