Adjoint matrix

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

Another way of writing for the adjoint matrix is <math> \ 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 <math> A< /math> one <math> n \ times n< /math> - matrix over the body <math> \ Bbb K< /math> the real or complex numbers, i.e. <math> \ Bbb K= \ R< /math> or <math> \ Bbb K= \ Bbb C< /math>.

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

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

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

One can show that the adjoint

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

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

Verallgemeinerung

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

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

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

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