Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Eigenvalues, eigenvectors, eigenspaces , and the spectrum of an endomorphism
Definition
Let be a vector space over a field and let be linear. A scalar is an eigenvalue of when there is a nonzero vector such that
Such a vector is an eigenvector belonging to . The eigenspace belonging to is
Thus is a linear subspace and always contains , while is an eigenvalue exactly when . The spectrum of over is
In particular, the unique endomorphism of the zero space has empty spectrum.
Depends on
Used by
- Algebraic multiplicity as the exponent of x-λ in χ_T, and geometric multiplicity as dim E_λ(T) Definition
- The matrix beginpmatrix2&10&3 endpmatrix has characteristic polynomial (x-2)(x-3) and two explicitly computed eigenspaces Example
- For a finite-dimensional space, λ is an eigenvalue of T if and only if T-λ I is not invertible Proposition
- Eigenvectors belonging to pairwise distinct eigenvalues are linearly independent Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 11 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)