Matriz transposta conjugada
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Em matemática, sobretudo na álgebra linear, uma matriz transposta conjugada de uma matriz n×m{displaystyle ntimes m,}
é uma outra matriz m×n{displaystyle mtimes n,}
formada pelo complexo conjugado de cada elemento da matriz transposta AT{displaystyle A^{T},}
. Costuma-se denotar por A∗{displaystyle A^{*},}
a matriz transposta conjugada de A{displaystyle A,}
: (A∗)i,j=Aj,i¯{displaystyle (A^{*})_{i,j}={overline {A_{j,i}}}}
, quando a matriz A{displaystyle A,} está escrita em uma base ortonormal.[1]
A matriz A∗{displaystyle A^{*},} representa o operador adjunto do operador linear associado à matriz A{displaystyle A,}. A propriedade fundamental do operador adjunto é dada pela igualdade:
- ⟨Ax,y⟩=⟨x,A∗y⟩,∀x,y{displaystyle langle Ax,yrangle =langle x,A^{*}yrangle ,forall x,y,}
Uma matriz é dita matriz hermitiana ou auto-adjunta se for idêntica à sua transposta conjugada.
Referências
↑ Horn, Roger A.; Johnson, Charles R. (1985). Matrix Analysis. [S.l.]: Cambridge University Press. ISBN 0-521-38632-2
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