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''of a linear transformation  $  A $''

The linear transformation  $  A  ^ {*} $
on a Euclidean space (or [[Unitary space|unitary space]])  $  L $
such that for all  $  x, y \in L $, 
the equality

$$ 
(Ax, y)  =  (x, A  ^ {*} y)
$$

between the scalar products holds. This is a special case of the concept of an adjoint linear mapping. The transformation  $  A  ^ {*} $
is defined uniquely by  $  A $. 
If  $  L $
is finite-dimensional, then every  $  A $
has an adjoint  $  A  ^ {*} $, 
the matrix  $  {\mathcal B} $
of which in a basis  $  e _ {1} \dots e _ {n} $
is related to the matrix  $  {\mathcal A} $
of  $  A $
in the same basis as follows:

$$ 
{\mathcal B}  =  \overline{G}\;  ^ {-1} {\mathcal A}  ^ {*} \overline{G}\; ,
$$

where  $  {\mathcal A}  ^ {*} $
is the matrix adjoint to  $  {\mathcal A} $
and  $  G $
is the [[Gram matrix|Gram matrix]] of the basis  $  e _ {1} \dots e _ {n} $.

In a Euclidean space,  $  A $
and  $  A  ^ {*} $
have the same characteristic polynomial, determinant, trace, and eigen values. In a unitary space, their characteristic polynomials, determinants, traces, and eigen values are [[complex conjugate]]s.

====Comments====
More generally, the phrase  "adjoint transformation"  or  "adjoint linear mappingadjoint linear mapping"  is also used to signify the dual linear mapping  $  \phi  ^ {*} :  M  ^ {*} \rightarrow L  ^ {*} $
of a linear mapping  $  \phi :  L \rightarrow M $. 
Here  $  M  ^ {*} $
is the space of (continuous) linear functionals on  $  M $
and  $  \phi  ^ {*} (m  ^ {*} (l)) = m  ^ {*} ( \phi (l)) $. 
The imbeddings  $  L \rightarrow L  ^ {*} $, 
$  M \rightarrow M  ^ {*} $, 
$  l \mapsto ( \cdot , l) $
connect the two notions. Cf. also [[Adjoint operator|Adjoint operator]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Reed,   B. Simon,   "Methods of modern mathematical physics" , '''1. Functional analysis''' , Acad. Press  (1972)  pp. Sect. 2</TD></TR></table>