Definition · applies to square matrices · also called diagonalization, spectral decomposition

Undergraduate

Consider a square matrix $A$. An eigendecomposition of $A$ is a factorization $A = P D P^{-1}$, where $P$ is invertible and $D$ is diagonal. The diagonal entries of $D$ are eigenvalues of $A$, and the columns of $P$ are corresponding eigenvectors.

Examples

Worked examples

  • The matrix \(\begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\) : \(P = \begin{bmatrix} 1 & 1 \\ \frac{1 + \sqrt{17}}{2} & \frac{1 - \sqrt{17}}{2} \end{bmatrix}\), $D = \operatorname{diag}\bigl(\frac{5 + \sqrt{17}}{2}, \frac{5 - \sqrt{17}}{2}\bigr)$ Undergraduate
    The matrix \(A = \begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\) has the distinct eigenvalues $\lambda = \frac{5 \pm \sqrt{17}}{2}$, with eigenvectors $(1, \lambda - 2)$, so it has an eigendecomposition. The eigenvectors are not perpendicular, because $A$ is not symmetric.
  • The matrix \(\begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}\) : $\Lambda = \operatorname{diag}\bigl(\frac{7 + \sqrt{17}}{2}, \frac{7 - \sqrt{17}}{2}\bigr)$, with $Q$ orthogonal Undergraduate
    The matrix \(A = \begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}\) is symmetric positive definite, so it has every decomposition in the table, including a Cholesky decomposition. Its eigendecomposition, Schur decomposition, Jordan normal form, and singular value decomposition all coincide, and its polar and Toeplitz decompositions are trivial.
  • The matrix \(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\) : None, since $A$ is not diagonalizable Undergraduate
    The matrix \(A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\) has no eigendecomposition, because its only eigenvalue is $1$ and its eigenvectors are the multiples of $(1, 0)$, so it does not have two independent eigenvectors.
  • The matrix \(A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\) : \(Q = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}\), $\Lambda = \operatorname{diag}(1, -1)$ Undergraduate
    The matrix \(A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\) is symmetric, so it has an eigendecomposition with an orthogonal $Q$: it fixes $(1, 1)$ and flips $(1, -1)$.

Properties

  • A square matrix $A$ has an eigendecomposition exactly when $A$ is diagonalizable, that is, when $A$ has $n$ linearly independent eigenvectors, where $A$ is $n \times n$. If the $n$ eigenvalues of $A$ are distinct, then $A$ is diagonalizable.
  • If $A$ is real and symmetric, then $A$ has an eigendecomposition $A = Q \Lambda Q^{\top}$ with $Q$ orthogonal and $\Lambda$ real (the spectral theorem).
  • An eigendecomposition turns powers and functions of $A$ into powers and functions of numbers: $A^k = P D^k P^{-1}$ and $e^{A t} = P e^{D t} P^{-1}$.

Notes

For a symmetric positive semidefinite matrix, the eigendecomposition $A = Q \Lambda Q^{\top}$ is also a singular value decomposition. For other matrices, the two differ: $P$ need not be orthogonal, and the entries of $D$ can be negative or complex.

Distinctions involving this term

  • Types of matrix decompositions: A matrix decomposition writes a matrix in terms of simpler matrices. Most are products: LU, PLU, and Cholesky decompositions use triangular factors, QR and singular value decompositions use orthogonal factors, and eigendecompositions, Schur decompositions, and Jordan normal forms are similarities $A = P M P^{-1}$ to a simpler matrix $M$ (diagonal, triangular, or made of Jordan blocks). In contrast, the Jordan–Chevalley decomposition is a sum of commuting diagonalizable and nilpotent parts, and the Toeplitz decomposition is a sum $H + K$ of a symmetric matrix $H$ and a skew-symmetric matrix $K$.

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