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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
- If W is T-invariant, then W^⊥ is T^*-invariant Lemma
- Adjoints satisfy (S+T)^*=S^*+T^*, (λ T)^*=overlineλ 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 · next 3 levels
Direct dependencies and their dependencies through the next three levels: 15 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
- 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)