Revision as of 06:35, 10 June 2024

Given a matrix, its eigenvectors are special vectors that satisfy the following property:

$A{\vec {x}}=\lambda {\vec {x}}$

where $\lambda$ is the eigenvalue associated with the eigenvector ${\vec {x}}$

The definition of eigenvectors are also frequently written in this form:

$(A-\lambda I){\vec {x}}=0$

Intuition

If we think of a matrix as a linear transformation, eigenvectors do not change direction. Instead, they simply scale by an eigenvalue.

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 [[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.