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.
The adjoint is characterised by
Definition
Let be inner product spaces (Real and complex inner product spaces, with the inner product linear in the first argument) over the same field and let be linear (Linear map between vector spaces over the same field). An adjoint of is a linear map such that
for every and . Existence is not part of the definition; in finite dimensions it will follow from Riesz representation.
Depends on
Used by
- Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space Definition
- If W is T-invariant, then W^⊥ is T^*-invariant Lemma
- Adjoints satisfy (S+T)^*=S^*+T^*, (λ T)^*=λ̄ T^*, (ST)^*=T^*S^*, and T^**=T Proposition
- An endomorphism is an orthogonal projection exactly when it is idempotent and self-adjoint Theorem
- Every linear map between finite-dimensional inner product spaces has a unique adjoint Theorem
- For a linear map T:V→ W between finite-dimensional inner-product spaces, x minimises ‖ Tx-b‖ if and only if T^*(Tx-b)=0, equivalently T^*Tx=T^*b; minimisers exist and any two differ by an element of ker T Theorem
- For an endomorphism in finite dimension, preserving lengths, preserving inner products, carrying orthonormal bases to orthonormal bases, and T^*T=I are equivalent Theorem
- In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix Theorem
- ker T^*=(imT)^⊥ and imT^*=(ker T)^⊥ in finite dimension Theorem
Dependency tree · two levels
6 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
- Sheldon Axler, Linear Algebra Done Right, 4th ed., definition 7.1 (standard reference, not scraped)
- Sergei Treil, Linear Algebra Done Wrong, §5.5.1 (standard reference, not scraped)