Vector-valued quadratic forms in control theory

For finite dimensional R-vector spaces U and V we consider a symmetric bilinear map B : U × U → V . This then defines a quadratic map QB : U → V by QB(u) = B(u, u). Corresponding to each λ ∈ V ∗ is a R-valued quadratic form λQB on U defined by λQB(u) = λ·QB(u). B is definite if there exists λ ∈ V ∗ so that λQB is positive-definite. B is indefinite if for each λ ∈ V \ann(image(QB)), λQB is neither positive nor negative-semidefinite, where ann denotes the annihilator.