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

• For split operators, Jordan blocks read off eigenspace multiplicities and both canonical polynomials Corollary
• Orthonormal eigenbasis for a compact self adjoint operator Corollary
• A defective Jordan block can split under perturbation at square-root scale 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
• Compatible left and right eigenvectors for a simple eigenvalue Definition
• Jordan blocks, Jordan strings, and their endpoints Definition
• Primary components ker q(T)ᵉ and generalised eigenspaces G_λ⁽ᵉ⁾(T)=ker(T-λ I)ᵉ Definition
• Ritz values and Ritz vectors extracted from the Arnoldi Hessenberg reduction Definition
• The matrix beginpmatrix2&10&3 endpmatrix has characteristic polynomial (x-2)(x-3) and two explicitly computed eigenspaces Example
• Volterra operator is Hilbert Schmidt and quasinilpotent Example
• Commuting endomorphisms preserve each other's eigenspaces Lemma
• Eigenspaces of a self adjoint operator are orthogonal Lemma
• Norm point of a compact self adjoint operator is an eigenvalue up to sign Lemma
• Orthogonal complement of an eigenspace is invariant Lemma
• Positive square root of a compact positive operator Lemma
• For a finite-dimensional space, λ is an eigenvalue of T if and only if T-λ I is not invertible Proposition
• An endomorphism is diagonalisable exactly when V=⨁_i<rE_λᵢ(T) for some finite list of distinct scalars λᵢ Theorem
• Eigenvectors belonging to pairwise distinct eigenvalues are linearly independent Theorem
• Every eigenvalue lies in some Gershgorin disk Theorem
• Singular value decomposition for compact operators Theorem
• Spectral theorem for compact self adjoint operators Theorem
• Trace of a positive operator is the sum of its eigenvalues Theorem

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)