Consider a closed convex cone \(C\subseteq\mathbb R^m\) defining on \(\mathbb R^m\) the usual cone order and thus a notion of positivity. Given a function \(f:\mathbb R^n\rightarrow \mathbb R^m\) and a domain \(D\subseteq \mathbb R^n,\) the question whether \(f(D)\subseteq C,\) i.e. whether \(f| D\) is nonnegative w.r.t. \(C\) can be difficult. For example, if \(m=n(n+1)/2,\) and \(\mathbb R^m\) parametrizes the set \(S^n\) of real symmetric \(n\times n\) matrices, we have the ordering defined by the cone of positive semidefinite matrices. Here, given \(A_0,A_1,\ldots, A_k \in S^n,\) the decision problem `is \(A_0+x_1A_1+\ldots+x_kA_k\succeq 0\) whenever \(\sum_i x_i^2\leq 1\)?' is known to be NP-complete; see \textit{A. Ben-Tal} and \textit{A. Nemirovski} [Math. Oper. Res. 23, No. 4, 769--805 (1998; Zbl 0977.90052)]. This paper shows that certain problems of this type involving matrices have nice solutions describable by linear matrix inequalities enlarging hereby the zoo of such examples; see e.g. \textit{Yu. Nesterov} [Appl. Optim. 33, 405--440 (2000; Zbl 0958.90090)] for earlier ones. Let \(C,B,A,D\) be real quadratic matrices. Using the S-lemma of \textit{V. A. Yakubovich} [5-procedure in nonlinear control theory. Vest. Leningr. Univ. 4, 73--93 (1977)], see \textit{J. F. Sturm} and \textit{S. Zhang} [``On cones of nonnegative quadratic functions'', Math. Oper. Res. 28, 246--267 (2003)], it is shown that there holds \(C+B^TX+XB+X^TAX \succeq 0\) whenever \(I+X^TDX\succeq 0\) iff for some \(t\geq 0\) there holds \[ \left[ \begin{matrix} C & B^T \\ B & A \end{matrix} \right]- t \left[ \begin{matrix} I & 0 \\ 0 & -D \end{matrix} \right] \succeq 0. \] Many more results of this type are deduced in section 3. In section 4 the setting is the following: Let \(L_{n,m}\) be the family of all \(nm\times nm\) matrices made from \(n^2\) matrices in \(S^m\) as `entries'. For \(D\subseteq \mathbb R^n, \) and \(\emptyset \neq\Delta \subseteq \mathbb R^m,\) let \(C_+(D,\Delta)=\{Z\in L_{n,m} : \sum_i \sum_j x_i x_j y^T Z_{ij} y \geq 0 \text{ for all } x\in D, y\in \Delta \};\) and let \(C_+(D)=C_+(D,\mathbb R)= \{Z\in S^n: x^T Z x\geq 0 \text{ for all }x\in D\)