Definition (the bilinear case).
Let be a finite-dimensional vector space over a field , and a symmetric, non-degenerate bilinear mapping, for example a real inner product. For an endomorphism we define the adjoint of relative to to be the endomorphism , characterized by
It is convenient to identify with a linear isomorphism in the sense that
We then have
To put it another way, gives an isomorphism between and the dual , and the adjoint is the endomorphism of that corresponds to the dual homomorphism (http://planetmath.org/DualHomomorphism) . Here is a commutative diagram to illustrate this idea:
Relation to the matrix transpose.
Let be a basis of , and let be the matrix of relative to this basis, i.e.
Let denote the matrix of the inner product relative to the same basis, i.e.
Then, the representing matrix of relative to the same basis is given by Specializing further, suppose that the basis in question is orthonormal, i.e. that
Then, the matrix of is simply the transpose .
The Hermitian (sesqui-linear) case.
If is an endomorphism of a unitary space (a complex vector space equipped with a Hermitian inner product (http://planetmath.org/HermitianForm)). In this setting we can define we define the Hermitian adjoint by means of the familiar adjointness condition
|Date of creation||2013-03-22 12:29:36|
|Last modified on||2013-03-22 12:29:36|
|Last modified by||rmilson (146)|