##### Actions

of a linear transformation $A$

The linear transformation $A ^ {*}$ on a Euclidean space (or 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 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 conjugates.

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.