Mathematics.

inner product spaces

Adjoint Operators

Linear Algebra45 minDifficulty7 out of 10

You should know: inner product spaces, linear transformation

Overview

The adjoint of a linear operator T on an inner product space is the unique operator T* satisfying ⟨Tu, v⟩ = ⟨u, T*v⟩ for all vectors u, v. Adjoints generalize the transpose (for real spaces) and the conjugate transpose (for complex spaces). Operators equal to their own adjoint (self-adjoint operators) have real eigenvalues and orthogonal eigenvectors.

Intuition

The adjoint T* is the operator you need to 'move T from one side of the inner product to the other'. In matrix terms, the adjoint of a real matrix is its transpose: ⟨Au, v⟩ = u·(Av) = (A^T u)·v = ⟨A^T u, v⟩. For complex matrices, it's the conjugate transpose A*. Self-adjoint operators (T = T*) are the linear-algebraic analogue of real numbers — they always have real eigenvalues.

Formal Definition

Definition

Let V be an inner product space and T: V → V a linear operator. The adjoint T* is the unique linear operator satisfying:

⟨Tu,v⟩=⟨u,T∗v⟩for all u,v∈V\langle Tu, v \rangle = \langle u, T^* v \rangle \quad \text{for all } u, v \in V
Defining property of the adjoint
For a real matrix A:A∗=A⊤\text{For a real matrix } A: \quad A^* = A^\top

The adjoint of a real matrix is its transpose.

Real case
For a complex matrix A:A∗=Aˉ⊤\text{For a complex matrix } A: \quad A^* = \bar{A}^\top

The adjoint of a complex matrix is its conjugate transpose.

Complex case

Notation

NotationMeaning
T∗T^*Adjoint of T (conjugate transpose for matrices)
A†A^\daggerConjugate transpose (physics notation for A*)
⟨u,v⟩\langle u,v \rangleInner product of u and v

Properties

Uniqueness

The adjoint is unique when it exists.\text{The adjoint is unique when it exists.}

Anti-linearity in composition

(ST)∗=T∗S∗(ST)^* = T^* S^*

Involution

(T∗)∗=T(T^*)^* = T

Self-adjoint (Hermitian)

T=T∗  ⟺  A=Aˉ⊤ (Hermitian matrix). Eigenvalues are real.T = T^* \iff A = \bar{A}^\top \text{ (Hermitian matrix). Eigenvalues are real.}

Skew-adjoint

T=−T∗  ⟺  A=−Aˉ⊤. Eigenvalues are purely imaginary.T = -T^* \iff A = -\bar{A}^\top. \text{ Eigenvalues are purely imaginary.}

Worked Examples

  1. 1

    For a real matrix with the standard inner product, the adjoint is the transpose.

    A∗=A⊤=(1234)A^* = A^\top = \begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix}
  2. 2

    Verify: ⟨Au, v⟩ = (Au)^T v = u^T A^T v = ⟨u, A^T v⟩ = ⟨u, A*v⟩. ✓

✓ Answer

A* = A^T = [[1,2],[3,4]].

Practice Problems

Mediumfree response

For A = [[1,2],[0,3]] over R with the standard inner product, find A* and determine whether A is self-adjoint.

Mediumfree response

Prove that if T is self-adjoint (T = T*) on a real inner product space, all eigenvalues of T are real.

Common Mistakes

Common Mistake

For complex matrices, the adjoint is just the transpose.

For complex matrices the adjoint (conjugate transpose) requires taking the complex conjugate of each entry AND transposing. Forgetting the conjugate leads to errors.

Quiz

The adjoint T* of a linear operator T is defined by:
For a complex matrix A, the adjoint A* equals:
A self-adjoint operator T = T* on a complex inner product space has:

Summary

  • The adjoint T* satisfies ⟨Tu,v⟩ = ⟨u,T*v⟩ for all u,v.
  • For real matrices, A* = A^T; for complex matrices, A* = conjugate transpose of A.
  • Self-adjoint (T = T*) operators have real eigenvalues and orthogonal eigenvectors.
  • Composition rule: (ST)* = T*S*; involution: (T*)* = T.
  • Skew-adjoint operators (T = -T*) have purely imaginary eigenvalues.

References

  1. WebsiteWikipedia — Hermitian adjoint