= Laplace-Lagrange inclination matrix
{c}
{title2=$B$}
For circular planetary orbits with <angular momenta> $L_j=M_j\sqrt{GM_\star a_j}$, the inclination matrix has positive off-diagonal entries, zero row sums, and $L_jB_{jk}=L_kB_{kj}$. <Positive diagonal symmetrization of a matrix> with $W=\operatorname{diag}(L_j)$ makes $W^{1/2}BW^{-1/2}$ a <symmetric matrix>. Moreover,
$$
\mathbf z^\dagger WB\mathbf z=-\sum_{j<k}L_jB_{jk}|z_j-z_k|^2\le0.
$$
All <eigenvalues> are real and nonpositive. The <zero eigenvalue> is a common tilt of every orbital plane; nonzero modes describe relative nodal precession.
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