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Adjungierte Matrix June 24, 2007

Filed under: Lineare Algebra — abi007 @ 12:03 pm

Adjungierte Matrix (conjugate matrix)

\begin{matrix} A*=A^H & = & \bar{A}^T \\ \bar{z} & = & a-b\cdot i \\ \langle Av, w\rangle & = & \langle v, A^*\,w\rangle, v, w \in\Bbb K^n \\ \Bbb K =\Bbb R & \rightarrow & A^*=A^T \\\end{matrix}

A=A* \Longleftrightarrow selbstadjungiert, symmetrisch; hermitesch

\begin{matrix}\left(A + B\right)^* &=& A^* + B^*\\rA)^* &=& \overline{r}A^*\\\left(AB\right)^* &=& B^*A^*\\\left(A^*\right)^* &=& A\\\left(A^{-1}\right)^* &=& \left(A^*\right)^{-1}\\det(A^*) &=&\overline{\det(A)}\\ trace(A^*) &=& \overline{trace(A)}\end{matrix}

adj(A) = \tilde A^T = \begin{pmatrix} \tilde a_{11} & \tilde a_{21} & \cdots & \tilde a_{n1}\\ \tilde a_{12} & \tilde a_{22} & & \tilde a_{n2}\\ \vdots & & \ddots & \vdots\\ \tilde a_{1n} & \tilde a_{2n} & \cdots & \tilde a_{nn}\end{pmatrix}

 

One Response to “Adjungierte Matrix”

  1. mathebuch Says:
    June 24, 2007 at 8:33 pm

    Wie kannst du die Formeln so gut in WordPress formatieren? Das sieht klasse aus.

    Reply

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