  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.
