It is proved that for any Lipschitz mapping T on the algebra M(n) of n x n matrices over the complex numbers satisfying T(0) = 0 and sigma(T(A) - T(B)) subset of sigma(A - B),A,B is an element of M(n), there exists an invertible matrix U is an element of M(n) such that T(A) = UAU(-1) for all A is an element of M(n) or T(A) = UA(t)U(-1) for all A is an element of M(n). [References: 4]
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