Matrize antisimetriko

Matrize antisimetrikoa (edo hemisimetrikoa) n×n elementuko matrize karratu bat da ${\displaystyle A=(a_{i,j})\in {\mathcal {M}}_{n\times n}}$, non ${\displaystyle i,j\in \{1,2,3,\dots ,n\}}$ guztietarako ${\displaystyle a_{i,j}=-a_{j,i}\,}$. Horren ondorioz, ${\displaystyle a_{ii}=0}$ i edozein izanda.

Hortaz, forma hau dauka:

${\displaystyle A={\begin{pmatrix}0&a_{12}&a_{13}&.&.&.&a_{1n}\\-a_{12}&0&a_{23}&.&.&.&a_{2n}\\-a_{13}&-a_{23}&0&.&.&.&a_{3n}\\.&.&.&.&.&.&.\\.&.&.&.&.&.&.\\.&.&.&.&.&.&.\\-a_{1n}&-a_{2n}&-a_{3n}&.&.&.&0\\\end{pmatrix}}}$

${\displaystyle A={\begin{pmatrix}{0}&{-2}&{4}\\{2}&{0}&{2}\\{-4}&{-2}&{0}\\\end{pmatrix}}}$ = > ${\displaystyle -A={\begin{pmatrix}{0}&{2}&{-4}\\{-2}&{0}&{-2}\\{4}&{2}&{0}\\\end{pmatrix}}}$

Ohar gaitezen A Matrize antisimetrikoaren matrize iraulia -A dela, eta antisimetria diagonal nagusiarekikoa dela.

n bakoitia bada, matrizearen determinante 0 da.