##  [Diagonalization](/diagonalization) 

  ##  [Diagonalization](https://mathlogic.quantumdictionary.io/diagonalization-0) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A general method that constructs a new object by systematically changing the diagonal components of a purported enumeration so the constructed object differs from every object in the list; used to prove nonenumerability, separations of hierarchies, or undecidability.

 

 

 

 

 





 

 



 ##  [Diagonalization](https://puremath.quantumdictionary.io/diagonalization-1) 

  

 [![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

The process of finding a basis in which a linear operator or matrix is represented by a diagonal matrix, equivalently finding a full set of eigenvectors so that the operator acts as scalar multiplication on each basis vector.

 

 

 

 

 





 

 



 ##  [Diagonalization](https://algebra.quantumdictionary.io/diagonalization-2) 

  

 [![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 process and property by which a linear operator or matrix is represented, relative to some basis, by a diagonal matrix; equivalently, the ambient vector space admits a basis made of eigenvectors for the operator.