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Mathematics for Physics

239 Adjoint

A linear transformation paired with another mapping through an inner product that is used to move that inner product from one factor to the other in an equation.

Note: Also called Hilbert-adjoint operator.

definition [d] (Adjoint = Hilbert-Adjoint Operator) From Kreyszig: the Hilbert-adjoint operator \(T^{*}: H \rightarrow H\) is defined to be the operator satisfying

  • \((Tx, y) = (x, T^{*}y)\)

for all \(x, y \in H\).

where

  • \(H\) is a Hilbert space.
  • \(T: H \rightarrow H\) is a given operator.
  • \(T^{*}\) is the adjoint of \(T\).
  • \((\cdot,\cdot)\) is the inner product on \(H\).

definition [d] (Adjoint) From Gowers, in the sense of Stokes’s theorem: one can view Stokes’s theorem as a definition of the derivative operation \(\omega \mapsto d\omega\); thus differentiation is the adjoint of the boundary operation.

where

  • \(d\) is the exterior derivative on forms.
  • \(\partial\) is the boundary operation on the region of integration.
  • Stokes’s theorem asserts \(\displaystyle \int_{S} d\omega = \int_{\partial S} \omega\).

239.1 Elementary Example

239.1.1 Simple

The adjoint \(T^{*}\) moves the inner product from \(Tx\) onto \(x\). On \(\mathbb{R}^{2}\) with the dot product, the adjoint of a matrix is its transpose.

\[ T = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix},\quad T^{*} = \begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix} \]

\[ (Tx,y) = (x, T^{*} y) \]

where

  • \(T\) is the operator.
  • \(T^{*}\) is the adjoint of \(T\).
  • \((\cdot,\cdot)\) is the inner product.

239.1.2 General

On \(\mathbb{C}^{3}\), the adjoint is the conjugate transpose.

\[ T = \begin{pmatrix} 1 & i & 0 \\ 0 & 2 & 1+i \\ 0 & 0 & 3 \end{pmatrix} \]

\[ T^{*} = \begin{pmatrix} 1 & 0 & 0 \\ -i & 2 & 0 \\ 0 & 1-i & 3 \end{pmatrix} \]

where

  • \(T^{*} = T^{\dagger}\) for matrices on \(\mathbb{C}^{n}\).

239.2 References

  1. Kreyszig, E. Introductory Functional Analysis with Applications. Wiley, 1989. — Hilbert-adjoint operator \(T^{*}\) with \((Tx,y)=(x,T^{*}y)\).
  2. Gowers, T., Barrow-Green, J., & Leader, I. (eds.). The Princeton Companion to Mathematics. Princeton University Press, 2008. — differentiation as the adjoint of the boundary operation.