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Mathematics for Physics

249 Eigenvalue

A scalar for which a nonzero vector is scaled by a linear transformation that is used to measure stretch along that special direction.

definition [d] (Eigenvalue) 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

  • \(Tv = \lambda v\) .

where

  • \(T\) is an operator on a vector space \(V\) over a field \(F\).
  • \(v\) is a nonzero vector in \(V\).
  • \(\lambda\) is the eigenvalue of \(T\).

definition [d] (Eigenvalue) From Cohen: when the eigenvalue equation

  • \(Av = \lambda v\)

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.

249.1 Elementary Example

249.1.1 Simple

A diagonal \(2 \times 2\) matrix has eigenvalues equal to its diagonal entries.

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

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

\[ \lambda = 2 \]

where

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

249.1.2 General

In three dimensions, a diagonal matrix has three eigenvalues on the diagonal.

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

\[ \lambda_{1} = 2,\quad \lambda_{2} = 3,\quad \lambda_{3} = 5 \]

\[ A e_{i} = \lambda_{i} e_{i} \]

where

  • \(\lambda_{1}, \lambda_{2}, \lambda_{3}\) are the eigenvalues of \(A\).
  • \(e_{1}, e_{2}, e_{3}\) are the standard basis eigenvectors.

249.2 References

  1. Axler, S. Linear Algebra Done Right. — \(\lambda \in F\) is an eigenvalue of \(T\) if \(Tv = \lambda v\) for some \(v \neq 0\).
  2. Cohen, M. X. Linear Algebra: Theory, Intuition, Code. Sincxpress BV, 2021. — eigenvalue equation \(Av = \lambda v\).