Konjugatu (matematika)

Matematikan, zenbaki konplexu baten konjugatua bere zati irudikariaren zeinua aldatuz lortzen da. Hortaz, zenbaki konplexu baten konjugatua

${\displaystyle z=a+ib\,}$

(non ${\displaystyle a}$ eta ${\displaystyle b}$ zenbaki errealak diren) hau da:

${\displaystyle {\overline {z}}=a-ib.\,}$

Konjugatua sarritan honela ere adierazten da: ${\displaystyle z^{*}}$

Propietateak

Biz ${\displaystyle (z,w)\in \mathbb {C} ^{2}}$.

• ${\displaystyle {\overline {z+w}}={\bar {z}}+{\bar {w}}}$

• ${\displaystyle {\overline {zw}}={\bar {z}}\times {\bar {w}}}$

• ${\displaystyle {\overline {\left({\frac {z}{w}}\right)}}={\frac {\bar {z}}{\bar {w}}}}$ (${\displaystyle w\,}$ nulua ez bada)

• ${\displaystyle \operatorname {Im} \left(z\right)=0}$ bada, orduan ${\displaystyle {\bar {z}}=z}$

• ${\displaystyle \left|{\bar {z}}\right|=\left|z\right|}$

• ${\displaystyle z{\overline {z}}=\left|z\right|^{2}}$

• ${\displaystyle z^{-1}={{\overline {z}} \over {\left|z\right|^{2}}}}$ (${\displaystyle z}$ nulua ez bada)

Kanpo estekak

• Datuak: Q381040
• Multimedia: Complex conjugation / Q381040