# Have orthogonal complex matrices appeared in the literature?

According to https://en.wikipedia.org/wiki/Orthogonal_matrix,
https://en.wikipedia.org/wiki/Unitary_matrix, and Friedberg et al.'s Linear Algebra (4th edition), a matrix $$A\in F^{n\times n}$$

• is unitary if $$A^*=A^{-1}$$ and
• is orthogonal if $$A^T=A^{-1}$$

no matter whether $$F$$ is $$\mathbb{C}$$ or $$\mathbb{R}$$. Therefore, if $$F=\mathbb{R}$$, then $$A$$ is unitary if and only if $$A$$ is orthogonal (since $$A^* = A^t$$). However, if $$F=\mathbb{C}$$, then a unitary matrix (e.g., the matrix consisting of a single entry $$i$$) is not necessarily orthogonal. I have seen a lot of unitary (i.e., orthogonal) real matrices and unitary complex matrices. But, I cannot find anything related to orthogonal complex matrices. Have orthogonal complex matrices appeared or been studied in the literature? If they have, can someone please provide a reference?

Also, if almost all orthogonal matrices in the literature are unitary, then are there any reasons why the definition of orthogonal matrix stays the current way? Wouldn't things be easier to define orthgonal matrix as unitary real matrix, just like orthogonal linear operator is defined as unitary linear operator on $$\mathbb{R}$$ (see https://en.wikipedia.org/wiki/Orthogonal_transformation, https://en.wikipedia.org/wiki/Unitary_transformation, and Friedberg et al.'s book)?

• I have seen that only once. Marvin Marcus mentioned it, this would be around 1970. I cannot now remeber any more about it. Dec 27 '19 at 21:38

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" split in terminology is useful in signaling whether real or complex spaces are involved, exactly because complex orthogonal matrices are rarely considered.
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 $$0$$.