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Mathematics for Physics

248 Eigendecomposition

A factorization of a matrix into eigenvectors and eigenvalues that is used to rewrite the matrix as diagonal in an eigenbasis.

Note: Also called diagonalization.

definition [d] (Eigendecomposition = Diagonalization) From Cohen: eigendecomposition writes

  • \(A = V \Lambda V^{-1}\) ,

so that matrix \(A\) is diagonal in basis \(V\). That is why eigendecomposition is also sometimes called diagonalization.

where

  • \(A\) is a square matrix that admits an eigenbasis.
  • \(V\) is a matrix whose columns are eigenvectors of \(A\).
  • \(\Lambda\) is the diagonal matrix of corresponding eigenvalues.
  • \(V^{-1}\) is the inverse of \(V\).

definition [d] (Diagonalizable) From Axler: an operator on \(V\) is called diagonalizable if the operator has a diagonal matrix with respect to some basis of \(V\).

where

  • \(V\) is a finite-dimensional vector space.
  • the basis that diagonalizes the operator is an eigenbasis when the diagonal entries are the eigenvalues.

248.1 Elementary Example

248.1.1 Simple

A diagonal matrix is already an eigendecomposition with \(V = I\).

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

\[ V = I,\quad \Lambda = A,\quad A = V \Lambda V^{-1} \]

where

  • \(V\) is the identity matrix.
  • \(\Lambda\) holds the eigenvalues \(2\) and \(3\).

248.1.2 General

In three dimensions, a diagonalizable matrix factors as \(V \Lambda V^{-1}\) with three independent eigenvectors.

\[ A = \operatorname{diag}(2,3,5) \]

\[ V = I_{3},\quad \Lambda = \operatorname{diag}(2,3,5) \]

\[ A = V \Lambda V^{-1} \]

where

  • \(I_{3}\) is the \(3 \times 3\) identity.
  • the columns of \(V\) are eigenvectors of \(A\).

248.2 References

  1. Cohen, M. X. Linear Algebra: Theory, Intuition, Code. Sincxpress BV, 2021. — \(A = V\Lambda V^{-1}\); eigendecomposition as diagonalization.
  2. Axler, S. Linear Algebra Done Right. — diagonalizable means a diagonal matrix in some basis.