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We show that the limiting eigenvalue density of the product of n identicallydistributed random matrices from an isotropic unitary ensemble (IUE) is equalto the eigenvalue density of n-th power of a single matrix from this ensemble,in the limit when the size of the matrix tends to infinity. Using thisobservation one can derive the limiting density of the product of n independentidentically distributed non-hermitian matrices with unitary invariant measures.In this paper we discuss two examples: the product of n Girko-Ginibre matricesand the product of n truncated unitary matrices. We also provide an evidencethat the result holds also for isotropic orthogonal ensembles (IOE).

New spectral relations between products and powers of isotropic random matrices