Foundations

Scientific relationships · 9 / 9

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

1. Eigenvalues and eigenvectors → Abstract vector spaces
   Uses definition
   Relationship sources · 1

   • Proof of the spectral theorem ↗

   Linked statements

   • For a linear operator T:V→V, a nonzero vector v is an eigenvector with eigenvalue λ when Tv=λv. The corresponding eigenspace consists of that equation's solutions, including zero.
2. Finite-dimensional self-adjoint spectral theorem → Eigenvalues and eigenvectors
   Uses definition
   Relationship sources · 1

   • Proof of the spectral theorem ↗

   Linked statements

   • A self-adjoint operator on a finite-dimensional real or complex inner-product space has real eigenvalues and an orthonormal basis of eigenvectors.
3. Real singular value decomposition → Eigenvalues and eigenvectors
   Uses definition
   Relationship sources · 1

   • Linear Algebra, Lecture 29: Singular value decomposition ↗

   Linked statements

   • Every real m×n matrix A admits A=UΣVᵀ with orthogonal U,V and rectangular diagonal Σ having nonnegative entries. These singular values are square roots of eigenvalues of AᵀA, with zero padding as needed.
4. Local equilibria and linearized stability → Eigenvalues and eigenvectors
   Uses definition
   Relationship sources · 1

   • Notes on Diffy Qs: Stability and classification of isolated critical points ↗

   Linked statements

   • In a C¹ planar autonomous system, equilibria are zeros of the vector field. If the Jacobian eigenvalues have nonzero real parts, negative real parts give a local attracting equilibrium, positive real parts give a source, and opposite-sign real eigenvalues give a saddle.
5. Power iteration with a dominant eigenvalue → Eigenvalues and eigenvectors
   Uses definition
   Relationship sources · 1

   • Fundamentals of Numerical Computation: Power iteration ↗

   Linked statements

   • For diagonalizable A with |λ₁|>|λ₂|≥… and a start having nonzero λ₁-eigenvector coefficient, repeated x_(k+1)=Ax_k/‖Ax_k‖₂ approaches the dominant eigendirection, up to phase or sign. Directional error is O((|λ₂|/|λ₁|)ᵏ); normalization limits overflow but does not supply the required spectral gap.
6. Diagonalization and eigenspace multiplicities → Eigenvalues and eigenvectors
   Uses definition
   Relationship sources · 1

   • Interactive Linear Algebra, section 5.4: Diagonalization ↗

   Linked statements

   • A real n×n matrix A is diagonalizable over ℝ exactly when it has n linearly independent real eigenvectors. With eigenvectors as columns of P, P⁻¹AP is diagonal with their corresponding eigenvalues. Distinct n real eigenvalues suffice; more generally the characteristic polynomial must split over ℝ and each eigenspace dimension equal its eigenvalue's algebraic multiplicity.
7. Cayley–Hamilton matrix identity → Eigenvalues and eigenvectors
   Uses definition
   Relationship sources · 1

   • Math220, section 6.A: Cayley–Hamilton theorem ↗

   Linked statements

   • For a finite square matrix A over a field, its characteristic polynomial p(t)=det(tI−A)=tⁿ+cₙ₋₁tⁿ⁻¹+⋯+c₀ satisfies p(A)=Aⁿ+cₙ₋₁Aⁿ⁻¹+⋯+c₀I=0. Polynomial division by p reduces any polynomial in A to degree below n; when c₀≠0 the identity also expresses A⁻¹ as a polynomial in A.
8. Euler–Lotka growth and age structure → Eigenvalues and eigenvectors
   Related to
   Relationship sources · 2

   • Lectures on Theoretical Ecology, §§5.1 and6.1: census conventions, survival and Euler–Lotka ↗
   • The evolution of life histories in spatially heterogeneous environments, equation1 and its homogeneous-environment limits ↗

   Linked statements

   • In a fixed density-independent age model, the growth factor λ satisfies Σ_{x≥1}λ^(−x)l_xm_x=1. For continuous age a, survival l(a) and daughter-production rate b(a), the corresponding exponential rate r satisfies ∫₀∞e^(−ra)l(a)b(a)da=1. Earlier reproduction can change growth even when expected lifetime offspring are unchanged.

   An age projection matrix connects asymptotic growth to its dominant eigenvalue.

9. Deficiency indices and self-adjoint extensions → Eigenvalues and eigenvectors
   Related to
   Relationship sources · 1

   • Introduction to spectral theory (2014) ↗

   Linked statements

   • For a closed densely defined symmetric operator T, deficiency indices n-plus and n-minus are the dimensions of ker(T*−iI) and ker(T*+iI). Self-adjoint extensions exist exactly when these dimensions agree, and are parameterized by unitary maps between deficiency spaces. For an initially nonclosed operator, both indices zero imply essential self-adjointness of its closure, rather than equality of the original domain with the adjoint domain.

   The kernels are eigenspaces at nonreal spectral parameters of the adjoint.