adjoint endomorphism

Definition (the bilinear case).

Let U be a finite-dimensional vector spaceMathworldPlanetmath over a field 𝕂, and B:U×U𝕂 a symmetricPlanetmathPlanetmathPlanetmathPlanetmath, non-degenerate bilinear mapping, for example a real inner productMathworldPlanetmath. For an endomorphismPlanetmathPlanetmath T:UU we define the adjointPlanetmathPlanetmathPlanetmath of T relative to B to be the endomorphism T:UU, characterized by


It is convenient to identify B with a linear isomorphism B:UU* in the sense that


We then have


To put it another way, B gives an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath between U and the dual U*, and the adjoint T is the endomorphism of U that corresponds to the dual homomorphism ( T*:U*U*. Here is a commutative diagramMathworldPlanetmath to illustrate this idea:


Relation to the matrix transpose.

Let 𝐮1,,𝐮n be a basis of U, and let MMatn,n(𝕂) be the matrix of T relative to this basis, i.e.


Let PMatn,n(𝕂) denote the matrix of the inner product relative to the same basis, i.e.


Then, the representing matrix of T relative to the same basis is given by P-1MtP. Specializing further, suppose that the basis in question is orthonormal, i.e. that


Then, the matrix of T is simply the transposeMathworldPlanetmath Mt.

The Hermitian (sesqui-linear) case.

If T:UU is an endomorphism of a unitary space (a complex vector space equipped with a Hermitian inner product ( In this setting we can define we define the Hermitian adjoint T:UU by means of the familiar adjointness condition


However, the analogous operationMathworldPlanetmath at the matrix level is the conjugate transposeMathworldPlanetmath. Thus, if MMatn,n() is the matrix of T relative to an orthonormal basisMathworldPlanetmath, then Mt¯ is the matrix of T relative to the same basis.

Title adjoint endomorphism
Canonical name AdjointEndomorphism
Date of creation 2013-03-22 12:29:36
Last modified on 2013-03-22 12:29:36
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 12
Author rmilson (146)
Entry type Definition
Classification msc 15A04
Classification msc 15A63
Synonym adjoint
Related topic Transpose
Defines Hermitian adjoint