Eigenvalues, eigenvectors, eigenspaces $E_{\lambda }(T)=\ker (T-\lambda I)$, and the spectrum ${\sigma }_F(T)$ of an endomorphism

Depends on

• Linear map between vector spaces over the same field
• Kernel and image of a linear map
• The space $\mathcal L(V,W)$ of linear maps with pointwise addition and scalar multiplication

Used by

• Finite-budget block LOBPCG for a Hermitian-definite pencil with explicit rank-loss status Algorithm
• Finite-order Padé scaling-and-squaring exponential Algorithm
• Shift-invert and harmonic Ritz extraction with solve and pencil status Algorithm
• For split operators, Jordan blocks read off eigenspace multiplicities and both canonical polynomials Corollary
• Non-invertible elliptic shifts form a discrete set in the self-adjoint case Corollary
• Orthonormal eigenbasis for a compact self adjoint operator Corollary
• The elliptic kernel and cokernel are finite dimensional Corollary
• A defective Jordan block can split under perturbation at square-root scale Counterexample
• Coercive non-symmetric forms need not have an orthonormal eigenbasis Counterexample
• The direct three-block ADMM extension can diverge Counterexample
• A diagonalisable endomorphism is one admitting a basis of eigenvectors, equivalently a diagonal matrix representation Definition
• Absolute value and singular values of a compact operator Definition
• Algebraic multiplicity as the exponent of x-λ in χ_T, and geometric multiplicity as dim E_λ(T) Definition
• Analytic primary matrix functions by Jordan blocks Definition
• Closed epsilon-pseudospectrum in the spectral norm Definition
• Compatible left and right eigenvectors for a simple eigenvalue Definition
• Hopf spectral and multilinear normal-form data Definition
• Jordan blocks, Jordan strings, and their endpoints Definition
• Local periodic-orbit, Poincare-return, and Floquet data Definition
• Primary components ker q(T)ᵉ and generalised eigenspaces G_λ⁽ᵉ⁾(T)=ker(T-λ I)ᵉ Definition
• Primary matrix sign for spectra off the imaginary axis Definition
• Real and complex Schur forms Definition
• Ritz values and Ritz vectors extracted from the Arnoldi Hessenberg reduction Definition
• Stabilizability and detectability for continuous and discrete linear systems Definition
• Symmetric elliptic weak eigenpairs Definition
• A coercive non-symmetric form can have non-real Galerkin eigenvalues Example
• A Jordan eigenpair residual can be much smaller than its eigenvalue error Example
• A repeated unit-cube Maxwell eigenvalue cluster Example
• A shift near the wanted eigenvalue suppresses its component under repeated implicit restarts Example
• A topological PEC zero mode on a spherical shell Example
• An explicit PEC standing wave in the unit cube Example
• An indefinite Coxeter form with a faithful canonical reflection representation Example
• The matrix beginpmatrix2&10&3 endpmatrix has characteristic polynomial (x-2)(x-3) and two explicitly computed eigenspaces Example
• The two-vertex deterministic swap has absolute gap zero Example
• Two-state reversible walk and its endpoint phases Example
• Volterra operator is Hilbert Schmidt and quasinilpotent Example
• A separated simple Floquet multiplier has a left/right-conditioned finite enclosure Lemma
• Commuting endomorphisms preserve each other's eigenspaces Lemma
• Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers Lemma
• Eigenspaces of a self adjoint operator are orthogonal Lemma

…and 28 more results.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• H. Pinkham, Linear Algebra, §§12.1–12.4 (standard reference, not scraped)
• M. Khovanov, Linear Algebra II notes, §6 (standard reference, not scraped)