[Math] On eigenvalues of complex-orthogonal matrices

linear algebramatrices

Suppose that $A$ is a complex matrix satisfying $A^TA = I$ (so $A$ is the entrywise transpose, not the conjugate transpose). What can be said about the eigenvalues of $A$, if $A$ is "complex-orthogonal" in this sense?

Of course, for any eigenpair $(\lambda,x)$, we have
$$
x^Tx = x^TA^TAx = (Ax)^TAx = \lambda^2 (x^Tx)
$$
which allows us to conclude that $\lambda^2 = 1$… so long as $x^Tx \neq 0$. Can anything else be said? Does the case in which $A$ has real entries allow us to conclude that $|\lambda| = 1$?

Best Answer

This has been thoroughly studied in the paper "The Jordan Canonical Forms of complex orthogonal and skew-symmetric matrices" by Horn and Merino (1999) and also in Olga Ruff's master thesis "The Jordan Canonical Forms of complex orthogonal and skew-symmetric matrices: characterisation and examples" (2007). In particular, theorem 1.2.3 (pp. 31-32) of Ruff's thesis states that

An $n\times n$ complex matrix is similar to a complex orthogonal matrix if and only if its Jordan Canonical Form can be expressed as a direct sum of matrices of only the following three types:

(a) $J_k(\lambda)\oplus J_k(\lambda^{-1})$ for $\lambda\in\mathbb C\setminus\{0\}$ and any $k$,

(b) $J_k(1)$ for any odd $k$ and

(c) $J_k(-1)$ for any odd $k$.

In particular, every nonzero complex number is an eigenvalue of some complex orthogonal matrix, and for each complex orthogonal matrix, all eigenvalues $\ne\pm1$ must occur in reciprocal pairs.

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