Definition · applies to square matrices and linear maps · also called additive Jordan decomposition, SN 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.

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.

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