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We consider level crossing in a matrix family $H=H_0+\lambda V$ where $H_0$is a fixed $N\times N$ matrix and $V$ belongs to one of the standard Gaussianrandom matrix ensembles. We study the probability distribution of levelcrossing points in the complex plane of $\lambda$, for which we obtain a numberof exact, asymptotic and approximate formulas.

Level Crossing in Random Matrices: I. Random perturbation of a fixed matrix