Two problems of the theory of quadratic maps

Given quadratic forms q (1), aEuro broken vertical bar, q (k) , two questions are studied: Under what conditions does the set of common zeros of these quadratic forms consist of the only point x = 0? When is the maximum of these quadratic forms nonnegative or positive for any x not equal 0? Criteria for each of these conditions to hold are obtained. These criteria are stated in terms of matrices determining the quadratic forms under consideration.