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  1. Linear Algebra
  2. Eigen-everything

Description

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Last updated 5 years ago

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An eigenvector of a linear transformation is a non-trivial vector that is only scaled when a linear transformation is applied to it. This means that the direction of the eigenvector is preserved after the operation.

In mathematical notation: Let v⃗\vec{v}v be an eigenvector of the matrix A\textbf{A}A. Then Av⃗=λv⃗\textbf{A}\vec{v} = \lambda\vec{v}Av=λv for a unique λ∈R\lambda \in \mathbb{R}λ∈R.

This equation says that the transformed vector Ax⃗\textbf{A}\vec{x}Ax is just a scaled version of the original vector v⃗\vec{v}v.

We also define the eigenvector to be non-trivial because A0⃗=0⃗\textbf{A}\vec{0} = \vec{0}A0=0 for all matrices.

The eigenvalue associated with this eigenvector is the scalar quantity by which the eigenvector is scaled. In the above equation, λ\lambdaλ would the eigenvalue associated with the vector v⃗\vec{v}v. The eigenvalue associated with an eigenvector can be 0.

The eigenspace of an eigenvalue is the span of all eigenvectors with this eigenvalue because any linear combination of the eigenvectors is still an eigenvector with the same eigenvalue. In the above equation the eigenspace Eλ=span{v⃗}E_{\lambda} = \text{span}\{\vec{v}\}Eλ​=span{v} because there is one eigenvector v⃗\vec{v}v corresponding to λ\lambdaλ.

Proof: Let v⃗\vec{v}v be an eigenvector of the matrix A\textbf{A}A. Then Av⃗=λv⃗\textbf{A}\vec{v} = \lambda \vec{v}Av=λv for a unique λ\lambdaλ. Now, let’s scale v⃗\vec{v}v by a constant ccc. A(cv⃗)=c(Av⃗)=c(λv⃗)=λ(cv⃗)\textbf{A}(c\vec{v}) = c(\textbf{A}\vec{v}) = c(\lambda \vec{v}) = \lambda (c\vec{v})A(cv)=c(Av)=c(λv)=λ(cv) Therefore, λ\lambdaλ is still the associated eigenvalue of cv⃗c\vec{v}cv.