Multiplicative maps on matrices that preserve the spectrum

Let Mn denote the set of all n × n matrices over the complex numbers ( n ≥ 2). Let An ⊆ Mn be either the set of all invertible matrices, the set of all unitary matrices, or a multiplicative semigroup containing the singular matrices. Theorem: If φ : An → Mn is a spectrum-preserving multiplicative map, then there exists a matrix R in Mn such that φ ( S ) = R −1 SR for all S in An .