Spectrum of a three-dimensional matrix with zero row sums

ConsiderIts row sums vanish, so $(1,1,1{)}^T$ is an eigenvector for the zero eigenvalue. Its characteristic polynomial iswith sum of principal minors of order twoThe other eigenvalues are $(-S\pm \sqrt{S^2-4T})/2$. No symmetry is required for this formula; positive diagonal symmetrization of a matrix is a sufficient condition for real roots.