# Adjoint linear transformation

*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.

#### 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.

#### References

[a1] | M. Reed, B. Simon, "Methods of modern mathematical physics" , 1. Functional analysis , Acad. Press (1972) pp. Sect. 2 |

**How to Cite This Entry:**

Adjoint linear transformation.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Adjoint_linear_transformation&oldid=45038