##  [Cayley–Hamilton Theorem](/cayley-hamilton-theorem) 

  ##  [Cayley–Hamilton Theorem](https://mathlogic.quantumdictionary.io/cayley-hamilton-theorem-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

Every square matrix over a commutative ring satisfies its own characteristic polynomial: if p(λ) = det(λI - A) is the characteristic polynomial of an n×n matrix A, then p(A) = 0 (the zero matrix), expressing a polynomial identity fulfilled by A.

 

 

 

 

 





 

 



 ##  [Cayley–Hamilton Theorem](https://puremath.quantumdictionary.io/cayley-hamilton-theorem-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 statement that every square matrix over a commutative ring (in particular over a field) satisfies its own characteristic polynomial: if p(λ) = det(λI − A) then p(A)=0 when p is evaluated with matrix substitution.

 

 

 

 

 





 

 



 ##  [Cayley–Hamilton Theorem](https://algebra.quantumdictionary.io/cayley-hamilton-theorem-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 statement that every square matrix satisfies its own characteristic polynomial: if p(λ) = det(λI − A) is the characteristic polynomial of a square matrix A, then p(A) = 0 (the zero matrix).