in linear algebra is to a real or complex square matrix [itex] A< /math> adjoint matrix [itex] A^*< /math> a matrix, which fulfills a certain permutation condition for dot products.

Another way of writing for the adjoint matrix is [itex] \ operator name {adj} (A)< /math>. This is to be enjoyed however with caution, thereit also for those, when Adjunkte designated, is used complementary matrix.

## definition

is [itex] A< /math> one [itex] n \ times n< /math> - matrix over the body [itex] \ Bbb K< /math> the real or complex numbers, i.e. [itex] \ Bbb K= \ R< /math> or [itex] \ Bbb K= \ Bbb C< /math>.

Too [itex] A< /math> adjoint matrix [itex] A^*< /math> is by the following characteristic defines:

[itex] \ langle Av, w \ rangle = \ langle v, A^* \, w \ rangle< /math> for all [itex] v, w \ in \ Bbb K^n< /math>.

Designation [itex] \ langle \ cdot, \ cdot \ rangle< /math> the canonical dot product [itex] \ of the Bbb K^n< /math>.

One can show that the adjoint

• in the real case [itex] \ Bbb K= \ R< /math> exactly the transponierte matrix< math> A^T< /math> by [itex] A< /math>is;
• in the complex case [itex] \ Bbb K= \ Bbb C< /math> exactly the complex conjugated the Transponierten, thus [itex] \ overline {A^T}< /math> is.

Math <if A^*> = A /math< applies>, then math <A> /math< is called> selbstadjungiert. In the complex case the matrix is called then also hermitesch.

## characteristics

• [itex] (A + B) ^* = A^* +B^*< /math> for two arbitrary stencils [itex] A< /math> and [itex] B< /math> with agreeing number of columns and line.
• [itex] (RA) ^* = r^*A^*< /math> for each complex number [itex] r< /math> and each matrix [itex] A< /math>. [itex] r^*< /math> here r /math stands for the number for <the complex> conjugated< math>.
• [itex] (OFF) ^* = B^*A^*< /math> for any [itex] (m, n)< /math> - matrix [itex] A< /math> andany [itex] (n, p)< /math> - matrix [itex] B< /math>.
• [itex] (A^*) ^* = A< /math> for any matrix [itex] A< /math>.
• [itex] \ langle Ax, y \ rangle = \ langle x, A^* y \ rangle< /math> for any [itex] (m, n)< /math> - matrix [itex] A< /math>, any vector [itex] x< /math> out [itex] \ mathbb {C} ^n< /math> and any vector [itex] y< /math> out [itex] \ mathbb {C} ^m< /math>. Math <\> langle designates. ,. \ rangle< /math> the usual Euclidean dot product (internal product) in [itex] \ mathbb {C} ^m< /math> and/or [itex] \ mathbb {C} ^n< /math>.

## Verallgemeinerung

in the Funktionalanalysis is generalized the adjoint matrix to adjoints the operator.

For a Endomorphismus [itex] F: V \ rightarrow V< /math> a Hilbertraums< math> V< /math> becomes a adjungierter Endomorphismus [itex] \ operator name {adj} (F): V \ rightarrow V< /math>by the characteristic:

[itex] \ langle \ operator name {F} (v), w \ rangle = \ langle v, \ operator name {adj} (F) (w) \ rangle< /math> for all [itex] v, w \ in \ V< /math>

defined. One knows then a connection to the binary operator [itex] F^*: V^* \ rightarrow V^*< /math> manufacture.