Adjunkte

the Adjunkte of a matrix, also classical adjoint or complementary matrix mentioned, is the transponierte matrix of the Kofaktoren, thus the signed Minoren. With the Adjunkten math A_ <{>i, j is said} to /math< briefly> from the matrix A the ith column and the j-width unit line painted. The sign changes, if i+j is odd.

the linguistic usage

the Adjunkte in today's text books emerges rarely there, and in older works the notation was everything else than clear, is required caution. Often the same notation for the Adjunkte and the adjoint (thus with real stencils their Transponierte, with complex stencils of them conjugate transponierte, with more general areas assistance of the which is the basis dot product or Sesquilinearproduktes) is used.

the classical use

the Adjunkte of <math> A \ in \ mathbb {C} ^ {n \ times n}< /math> in the classical sense A is given <}> = \ mbox {adj} (A) /math as math \ tilde< {> with elements

   <math> \ tilde {A} _ {ij} = (- 1) ^ {i+j}       \ begin {vmatrix}         a_ {1.1} & \ cdots & a_ {1, i-1} & a_ {1, i+1} & \ cdots & a_ {1, n} \ \         \ vdots & \ ddots & \ vdots & \ vdots & \ ddots & \ vdots \ \         a_ {j-1,1} & \ cdots & a_ {j-1, i-1} & a_ {j-1, i+1} & \ cdots & a_ {j-1, n} \ \         a_ {j+1,1} & \ cdots & a_ {j+1, i-1} & a_ {j+1, i+1} & \ cdots & a_ {j+1, n} \ \         \ vdots & \ ddots & \ vdots & \ vdots & \ ddots & \ vdots \ \         a_ {n, 1} & \ cdots & a_ {n, i-1} & a_ {1, i+1} & \ cdots & a_ {n, n}       \ end {vmatrix}. </math>

From the definition of the Adjunkten and the definition of the determinants after Leibniz one sees direct that the formula of Cauchy - Binet applies:

 <math> \ mbox {adj} (A) A=A \ mbox {adj} (A)= \ December (A) Ith< /math>

This formula forms the basis of the definition only for <math> \ December (A) \ of the neq0< /math> existing and then clear (left and on the right of) inverse ones as

  <math> A^ {- 1} = \ frac {\ mbox {adj} (A)} {\ December (A)}. </math>

Over the connection between the Resolvente, i.e., to which inverse ones of the spectrally shifted matrix and self-vectors leave themselves statements about the self-vectors of stencils to win.