Given a linear operator on an inner product space, we can define its “adjoint” by transposing the role of the two arguments in the inner product.
Definition 9.3.1 (Adjoint operator). Let be a finite-dimensional inner product space and a linear operator. The adjoint of is the unique linear operator satisfying for all .
Theorem 9.3.1 (Existence and uniqueness of the adjoint). For any linear operator on a finite-dimensional inner product space, the adjoint exists and is unique.
Proof (Existence). Fix . The map is a linear functional on .
Since the inner product is nondegenerate, the map defined by is an isomorphism. Thus there exists a unique with for all . Define .
We must check is linear. For and : Since this holds for all , we get .
Uniqueness: If also satisfies , then for all , so by nondegeneracy. ∎
Note. Alternatively, the adjoint can be described as follows. The inner product gives an isomorphism ,. The transpose is defined by . Then This shows the adjoint is the “conjugate” of the transpose by the inner-product isomorphism.
9.3.2 Properties of the Adjoint
Proposition 9.3.1 (Algebraic properties of the adjoint). Let be linear operators on an inner product space, and . Then:
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Proof. Each follows from manipulating the defining equation .
(i) . So .
(iv) .∎
Note. Property (iv) shows that taking adjoints reverses order, just like taking transposes or inverses. The map is a conjugate-linear anti-automorphism of .
Matrix of the Adjoint
Theorem 9.3.2 (Matrix of the adjoint). Let be an orthonormal basis of , and let be the matrix of in this basis. Then the matrix of is: where denotes the conjugate transpose.
Proof. The -entry of is the coefficient of in , which equals .
The -entry of is .
Using the defining property: The -entry of is , so ∎
Note. In an orthonormal basis, self-adjoint operators () correspond to symmetric matrices (real case) or Hermitian matrices (complex case, ).
Proposition 9.3.2 (Kernel and image of the adjoint). For :
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Proof. (i) .
(ii) Apply (i) to :, i.e., . Taking orthogonal complements: (using in finite dimensions). ∎