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We consider the limiting location and limiting distribution of the largesteigenvalue in real symmetric ($\beta$ = 1), Hermitian ($\beta$ = 2), andHermitian self-dual ($\beta$ = 4) random matrix models with rank 1 externalsource. They are analyzed in a uniform way by a contour integral representationof the joint probability density function of eigenvalues. Assuming the one-bandcondition and certain regularities of the potential function, we obtain thelimiting location of the largest eigenvalue when the nonzero eigenvalue of theexternal source matrix is not the critical value, and further obtain thelimiting distribution of the largest eigenvalue when the nonzero eigenvalue ofthe external source matrix is greater than the critical value. When the nonzeroeigenvalue of the external source matrix is less than or equal to the criticalvalue, the limiting distribution of the largest eigenvalue will be analyzed ina subsequent paper. In this paper we also give a definition of the externalsource model for all $\beta$ > 0.

The largest eigenvalue of real symmetric, Hermitian and Hermitian self-dual random matrix models with rank one external source, part I