One can apply the Hahn-Jordan decomposition to these measures to split them as
2.
This consequence of the Hahn decomposition theorem is called the " Jordan decomposition ".
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To give some idea of this generalization, we first reformulate the Jordan decomposition in the language of functional analysis.
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The Jordan decomposition in measure theory says that every signed measure can be expressed as the difference of two positive measures supported on disjoint sets.
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If is a matrix with positive eigenvalues and minimal polynomial, then the Jordan decomposition into generalized eigenspaces of can be deduced from the partial fraction expansion of.
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For example, the Jordan decomposition in a semisimple Lie algebra coincides with the Jordan decomposition in its representation; this is not the case for Lie algebras in general.
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For example, the Jordan decomposition in a semisimple Lie algebra coincides with the Jordan decomposition in its representation; this is not the case for Lie algebras in general.
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Using generalized eigenvectors, we can obtain the Jordan normal form for A and these results can be generalized to a straightforward method for computing functions of nondiagonalizable matrices . ( See Matrix function # Jordan decomposition .)
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An arbitrary irreducible character has a " Jordan decomposition " : to it one can associate a semisimple character ( corresponding to some semisimple element " s " of the dual group ), and a unipotent representation of the centralizer of " s ".
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It can be proved that a real function " ? " is of bounded variation in [ a, b ] if and only if it can be written as the difference " ? " = " ? " 1 " " ? " 2 of two non-decreasing functions on [ a, b ] : this result is known as the Jordan decomposition of a function and it is related to the Jordan decomposition of a measure.
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