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Homomorphisms from the unitary group to the general linear group over complex number field and applications

Chong-Guang Cao;  Xian Zhang — 2002

Archivum Mathematicum

Let M n be the multiplicative semigroup of all n × n complex matrices, and let U n and G L n be the n –degree unitary group and general linear group over complex number field, respectively. We characterize group homomorphisms from U n to G L m when n > m ≥ 1 or n = m ≥ 3 , and thereby determine multiplicative homomorphisms from U n to M m when n > m ≥ 1 or n = m ≥ 3 . This generalize Hochwald’s result in [Lin. Alg. Appl.  212/213:339-351(1994)]: if f : U n → M n is a spectrum–preserving multiplicative homomorphism, then there exists a matrix R in G L n such that f ( A ) = R A R for...

Matrix inequalities involving the Khatri-Rao product

Xian Zhang;  Zhong Peng Yang;  Chong-Guang Cao — 2002

Archivum Mathematicum

We extend three inequalities involving the Hadamard product in three ways. First, the results are extended to any partitioned blocks Hermitian matrices. Second, the Hadamard product is replaced by the Khatri-Rao product. Third, the necessary and sufficient conditions under which equalities occur are presented. Thereby, we generalize two inequalities involving the Khatri–Rao product.

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