  In this paper we study perturbed Ornstein-Uhlenbeck operators \begin{align*}
\left[ \mathcal{L}_{\infty} v\right](x) = A\triangle v(x) + \left\langle
Sx,\nabla v(x)\right\rangle-B v(x),\,x\in\mathbb{R}^d,\,d\geqslant 2,
\end{align*} for simultaneously diagonalizable matrices
$A,B\in\mathbb{C}^{N,N}$. The unbounded drift term is defined by a
skew-symmetric matrix $S\in\mathbb{R}^{d,d}$. Differential operators of this
form appear when investigating rotating waves in time-dependent reaction
diffusion systems. As shown in a companion paper, one key assumption to prove
resolvent estimates of $\mathcal{L}_{\infty}$ in
$L^p(\mathbb{R}^d,\mathbb{C}^N)$, $1<p<\infty$, is the following
$L^p$-dissipativity condition \begin{align*} |z|^2\mathrm{Re} \left\langle w,Aw
\right\rangle + (p-2)\mathrm{Re} \left\langle w,z \right\rangle\mathrm{Re}
\left\langle z,Aw \right\rangle \geqslant \gamma_A |z|^2|w|^2\; \forall\, z,w
\in \mathbb{C}^N \end{align*} for some $\gamma_A>0$. We prove that the
$L^p$-dissipativity condition is equivalent to a new $L^p$-antieigenvalue
condition \begin{align*} A\text{ invertible} \quad \text{and} \quad \mu_1(A) >
\frac{|p-2|}{p}, \,1<p<\infty, \,\mu_1(A) \text{ first antieigenvalue of $A$,}
\end{align*} which is a lower $p$-dependent bound of the first antieigenvalue
of the diffusion matrix $A$. This relation provides a complete algebraic
characterization and a geometric meaning of $L^p$-dissipativity for
complex-valued Ornstein-Uhlenbeck operators in terms of the antieigenvalues of
$A$. The proof is based on the method of Lagrange multipliers. We also discuss
several special cases in which the first antieigenvalue can be given
explicitly.
