2022/07/02 by Guo, Feng, Yan, Sizhuo, Zhi, Lihong
#FOS: Mathematics #Operator Algebras (math.OA) #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2207.00840
We consider the problem of extending the classical S-lemma from commutative case to noncommutative cases. We show that a symmetric quadratic homogeneous matrix-valued polynomial is positive semidefinite if and only if its coefficient matrix is positive semidefinite. Then we extend the S-lemma to three kinds of noncommutative polynomials: noncommutative polynomials whose coefficients are real numbers, matrix-valued noncommutative polynomials and hereditary polynomials.