Jordan decomposition

The decomposition of a linear operator into a sum of commuting semisimple and nilpotent parts. In finite-dimensional settings, these parts are uniquely determined under standard field assumptions.

Jordan–Chevalley decomposition: A decomposition of a linear operator into commuting semisimple and nilpotent parts. This is the standard name for the additive decomposition described here.
Linear operator: A linear map from a vector space to itself. Jordan decomposition studies how one such map splits into simpler commuting maps.
Camille Jordan: A French mathematician known for work in group theory and the canonical forms of linear transformations. Jordan's work on canonical forms supplied the matrix setting behind the decomposition's name.
Matrix exponential: The operator-valued power series obtained by applying the exponential function to a square matrix. Commuting parts let the exponential factor into a semisimple contribution and a nilpotent polynomial factor.
Polar decomposition: A factorization of a linear operator into a unitary factor and a positive semidefinite factor. Unlike Jordan decomposition, it separates geometric isometry from magnitude rather than semisimple from nilpotent behavior.
Jordan normal form: A canonical matrix form built from blocks with one eigenvalue on the diagonal and ones on the superdiagonal. Its blocks reveal the semisimple and nilpotent contributions directly.
Eigenvalue: A scalar by which a linear operator scales some nonzero vector. Eigenvalues determine the semisimple action and the generalized eigenspaces.
Gaston Chevalley: A French mathematician who made major contributions to algebraic groups and Lie theory. Chevalley's name is attached to the decomposition's structural formulation.
Linear differential equation: A differential equation in which the unknown function and its derivatives occur linearly. For constant-coefficient systems, the decomposition clarifies exponential growth and polynomial factors in solutions.
Singular value decomposition: A matrix factorization into two unitary matrices and a diagonal matrix of nonnegative singular values. It applies to arbitrary matrices and does not require commuting semisimple and nilpotent summands.

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