##  [Eigenvalue](/eigenvalue-0) 

 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.

 

 

 

 

 

 





## Principle

Principle

Eigenvalues identify directions (eigenvectors) that are invariant up to scaling under a linear transformation; the characteristic polynomial det(A − λI)=0 yields the eigenvalues.

 

 

 

 

 





## Demonstration

Demonstration

For diagonal matrix diag(2,3), the eigenvalues are 2 and 3 with eigenvectors aligned with the coordinate axes; for a Jacobian matrix at a fixed point, eigenvalues determine local stability.

 

 

 

 

## Misapplication

Misapplication

Asserting eigenvectors exist for arbitrary nonlinear maps without linearization, or treating eigenvalues as unique scalar diagnostics without considering multiplicity, complex values, or defective matrices.

 

 

 

 

 





## Consequence

Consequence

Eigenvalues enable modal decomposition, stability and resonance analysis, dimensionality reduction (PCA uses eigenvectors of covariance), and spectral representations in differential operators.

 

 

 

 

## Reversal

Reversal

Singular values from the singular value decomposition, which measure input–output gain without requiring square matrices or invariant directions, emphasizing magnitude rather than signed scaling and possibly differing from eigenvalues.

 

 

 

 

 





## Boundary

Boundary

Defined for linear operators on vector spaces (finite or infinite dimensional) and their matrix representations; excludes direct application to inherently nonlinear dynamics except via linearization and excludes non-spectral invariants.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Often conflated with singular values or roots of characteristic polynomials generically; eigenvalues are specific spectral scalars tied to invariant directions and depend on field (real/complex) and operator domain.

 

 

 

 

 





## Synthesis

Synthesis

An eigenvalue is a spectral scalar λ such that a linear transformation scales some nonzero vector v by λ; together with eigenvectors it decomposes linear action into invariant modes used across analysis, stability, and dimensionality reduction.