# Adjoint linear transformation

*of a linear transformation *

The linear transformation on a Euclidean space (or unitary space) such that for all , the equality

between the scalar products holds. This is a special case of the concept of an adjoint linear mapping. The transformation is defined uniquely by . If is finite-dimensional, then every has an adjoint , the matrix of which in a basis is related to the matrix of in the same basis as follows:

where is the matrix adjoint to and is the Gram matrix of the basis .

In a Euclidean space, and 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 of a linear mapping . Here is the space of (continuous) linear functionals on and . The imbeddings , , 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://www.encyclopediaofmath.org/index.php?title=Adjoint_linear_transformation&oldid=35199