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The adjoint of an operator

Riesz hands us the adjoint T* — the operator that moves T across the inner product. Learn its defining equation, compute it as a conjugate transpose, and master the algebra that every spectral theorem leans on.

Defining T* through pairing

Given a linear operator T on V, fix a vector w and consider v -> <Tv, w>. That is a linear functional in v, so Riesz produces a unique vector — call it T*w. The adjoint operator T* is defined by the single identity <Tv, w> = <v, T*w> for all v, w. It is the operator that lets you slide T from one side of the inner product to the other.

Computing it: the conjugate transpose

In an orthonormal basis the abstract definition becomes concrete: the matrix of T* is the conjugate transpose of the matrix of T, written A* = conj(A)^T (also A^H). Over R this is just the transpose A^T; over C you transpose and conjugate every entry. The conjugation is exactly the price the Hermitian inner product charged us in guide one.

A = [ 1+i,   2 ]
    [   3,  4-i ]

Conjugate transpose A* = conj(A)^T:

step 1  transpose:        [ 1+i,   3  ]
                          [   2,  4-i ]

step 2  conjugate each:   A* = [ 1-i,   3  ]
                               [   2,  4+i ]

Sanity check the defining identity on e1, e2 (standard inner product):
  <A e1, e2> = (A)_{21} = 3        (entry row 2, col 1)
  <e1, A* e2> = conj( (A*)_{12} ) = conj(3) = 3   -> they match.
Over C the adjoint transposes and conjugates; over R it is the plain transpose.

The algebra you must internalize

The adjoint identities turn it into a tidy calculus. The starred map reverses products and is conjugate-linear in scalars, while being an involution:

(S + T)*  = S* + T*
(c T)*    = conj(c) T*        (conjugation appears over C)
(S T)*    = T* S*             (order reverses, like the transpose)
(T*)*     = T                 (an involution: starring twice undoes itself)
(T^-1)*   = (T*)^-1           (when T is invertible)
Five identities that the adjoint always obeys.

One structural gem connects the adjoint to last guide's geometry: ker(T*) = (im T)-perp, the kernel of the adjoint is the orthogonal complement of the image. This is the conceptual upgrade of Vol I's "row space and null space are orthogonal" — the four fundamental subspaces, now stated operator-to-operator without ever choosing coordinates.