##  [Jordan Decomposition](/jordan-decomposition) 

  ##  [Jordan Decomposition](https://puremath.quantumdictionary.io/jordan-decomposition-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A procedure that expresses a linear endomorphism of a finite-dimensional vector space (over a suitable field) as a direct sum of Jordan blocks, or equivalently decomposes an endomorphism into the commuting sum of a semisimple (diagonalisable) part and a nilpotent part that share the same generalized eigenspaces.

 

 

 

 

 





 

 



 ##  [Jordan Decomposition](https://algebra.quantumdictionary.io/jordan-decomposition-1) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The canonical splitting of a linear endomorphism into a sum of commuting parts: a semisimple (diagonalizable) part and a nilpotent part, each a polynomial in the original operator; the decomposition is unique when it exists.