# Tag Info

Yes, orthogonal matrices with complex entries appeared at least as early as 1900, in E. Cartan's classification of simple Lie algebras (and Lie groups). In many ways, the complex numbers could be replaced by any algebraically closed field of characteristic $0$. Thinking of complex orthogonal groups as real Lie groups ("forgetting" the complex structure) ...
They did. A natural way to treat such matrices is to introduce an indefinite inner product on $\mathbb{C}^n$, a non-degenerate bilinear form $(z,w):=z_1w_1+z_2w_2+\dots z_nw_n$, instead of the usual sesquilinear one. Then $A^T=A^{-1}$ is equivalent to $(Az,Aw)=(z,w)$, i.e. complex orthogonal matrices are isometries of this space. The "orthogonal"/"unitary" ...