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Mathematics for Physics

250 Eigenvector

A nonzero vector that a linear transformation maps to a scalar multiple of itself that is used to identify special directions of that transformation.

definition [d] (Eigenvector) From Axler: a number \(\lambda \in F\) is called an eigenvalue of \(T\) if there exists \(v \in V\) such that \(v \neq 0\) and \(Tv = \lambda v\). A vector \(v \in V\) with \(v \neq 0\) is an eigenvector of \(T\) corresponding to \(\lambda\) if and only if

  • \(v \in \operatorname{null}(T - \lambda I)\) .

where

  • \(T\) is an operator on a vector space \(V\) over a field \(F\).
  • \(v\) is a nonzero eigenvector.
  • \(\lambda\) is the corresponding eigenvalue.
  • \(I\) is the identity operator on \(V\).
  • \(\operatorname{null}(T - \lambda I)\) is the null space of \(T - \lambda I\).

definition [d] (Eigenvector) From Cohen: when

  • \(Av = \lambda v\)

is satisfied, then \(v\) is an eigenvector and \(\lambda\) is its associated eigenvalue.

where

  • \(A\) is a square matrix.
  • \(v\) is a nonzero vector.
  • \(\lambda\) is the associated eigenvalue.

250.1 Elementary Example

250.1.1 Simple

On \(\mathbb{R}^{2}\), a diagonal matrix stretches the axis vectors by the diagonal entries.

\[ A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} \]

\[ v = \begin{pmatrix} 1 \\ 0 \end{pmatrix},\quad Av = \begin{pmatrix} 2 \\ 0 \end{pmatrix} = 2 v \]

where

  • \(v\) is an eigenvector of \(A\).
  • \(\lambda = 2\) is the corresponding eigenvalue.

250.1.2 General

In three dimensions, each standard basis vector can be an eigenvector of a diagonal matrix.

\[ A = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{pmatrix} \]

\[ v_{1} = e_{1},\quad A v_{1} = 2 v_{1} \]

\[ v_{2} = e_{2},\quad A v_{2} = 3 v_{2} \]

\[ v_{3} = e_{3},\quad A v_{3} = 5 v_{3} \]

where

  • \(e_{1}, e_{2}, e_{3}\) are the standard basis vectors of \(\mathbb{R}^{3}\).
  • each \(v_{i}\) is a nonzero eigenvector of \(A\).

250.2 References

  1. Axler, S. Linear Algebra Done Right. — eigenvector \(v \neq 0\) for \(\lambda\) means \(Tv = \lambda v\), equivalently \(v \in \operatorname{null}(T - \lambda I)\).
  2. Cohen, M. X. Linear Algebra: Theory, Intuition, Code. Sincxpress BV, 2021. — eigenvalue equation \(Av = \lambda v\).