Jordan–Chevalley decomposition
Undergraduate (complex matrices)
The Jordan–Chevalley decomposition of a square matrix $A$ writes it as a sum $A = D + N$, where $D$ is diagonalizable, $N$ is nilpotent ($N^k = 0$ for some $k$), and $D N = N D$.
Graduate (linear maps over a perfect field)
The Jordan–Chevalley decomposition of a linear map $T$ on a finite-dimensional vector space writes it as $T = S + N$, where $S$ is semisimple, $N$ is nilpotent, and $S$ and $N$ commute.
Examples
Worked examples
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The matrix \(\begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\)
: $D = A$, $N = 0$
Undergraduate
The matrix \(A = \begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\) has distinct eigenvalues, so it is diagonalizable. Thus its Jordan–Chevalley decomposition has $D = A$ and $N = 0$.
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The matrix \(\begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}\)
: $D = A$, $N = 0$
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.
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The matrix \(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\)
: \(D = I\), \(N = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\)
Undergraduate
The matrix \(A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\) is a Jordan block, which is not diagonalizable, so its Jordan–Chevalley decomposition has a nonzero nilpotent part $N = A - I$. $I$ commutes with every matrix, so $D = I$ and $N$ commute.
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The matrix \(A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)
: $D = A$, $N = 0$
Undergraduate
$A$ is orthogonal, so its QR decomposition is $A \cdot I$, its singular value decomposition is $A \cdot I \cdot I$, and its Jordan–Chevalley decomposition has $N = 0$ because $A$ is diagonalizable.
Properties
- The decomposition exists and is unique. Both $D$ and $N$ are polynomials in $A$.
- In a Jordan normal form, $D$ is the diagonal part and $N$ is the part above the diagonal.
Notes
Unlike the LU, QR, and singular value decompositions, the Jordan–Chevalley decomposition is a sum, not a product.
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 $K$.