##  [Eigenvalue](/eigenvalue) 

  ##  [Eigenvalue](https://quantummech.quantumdictionary.io/eigenvalue-1) 

  

 [![Quantum Mechanics Dictionary](/sites/default/files/styles/large/public/2026-01/Quantum%20Mechanics.png.webp?itok=tC_FGCW0)](/topic-specific-dictionaries/modern-physics/quantum-mechanics)

- Modern Physics -

**Quantum Mechanics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A mathematical structure used to represent quantum states and the operations performed on them. It specifies how states are expressed, related, combined, and decomposed into components that support calculation. It does not by itself determine physical predictions without a mapping to observables and measurement rules.



 

 

 

 

 





 

 



 ##  [Eigenvalue](https://natural.quantumdictionary.io/eigenvalue-0) 

  

 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/topic-specific-dictionaries/natural-formal-sciences)



**Natural & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A scalar λ such that for a linear operator or square matrix A there exists a nonzero vector v with Av = λv; λ belongs to the spectrum of A and characterizes invariant scaling along v.

 

 

 

 

 





 

 



 ##  [Eigenvalue](https://mathlogic.quantumdictionary.io/eigenvalue-2) 

  

 [![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

An eigenvalue of a linear operator A on a vector space over a field is a scalar λ for which there exists a nonzero vector v (an eigenvector) satisfying A v = λ v; equivalently λ lies in the point spectrum of A when A−λI fails to be injective.

 

 

 

 

 





 

 



 ##  [Eigenvalue](https://puremath.quantumdictionary.io/eigenvalue-3) 

  

 [![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 scalar λ for which there exists a nonzero vector v (an eigenvector) satisfying (T−λI)v=0 for a linear operator T, indicating a direction that is invariant up to scaling by λ.

 

 

 

 

 





 

 



 ##  [Eigenvalue](https://algebra.quantumdictionary.io/eigenvalue-4) 

  

 [![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

A scalar λ for which there exists a nonzero vector v satisfying Av = λv for a linear operator or matrix A; equivalently λ is in the point spectrum of A.