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Following our recent letter, we study in detail an entry-wise diffusion ofnon-hermitian complex matrices. We obtain an exact partial differentialequation (valid for any matrix size $N$ and arbitrary initial conditions) forevolution of the averaged extended characteristic polynomial. The logarithm ofthis polynomial has an interpretation of a potential which generates a Burgersdynamics in quaternionic space. The dynamics of the ensemble in the large $N$is completely determined by the coevolution of the spectral density and acertain eigenvector correlation function. This coevolution is best visible inan electrostatic potential of a quaternionic argument built of two complexvariables, the first of which governs standard spectral properties while thesecond unravels the hidden dynamics of eigenvector correlation function. Weobtain general large $N$ formulas for both spectral density and 1-pointeigenvector correlation function valid for any initial conditions. We exemplifyour studies by solving three examples, and we verify the analytic form of oursolutions with numerical simulations.

Unveiling the significance of eigenvectors in diffusing non-hermitian matrices by identifying the underlying Burgers dynamics