In my work in PDE, the following problem in linear algebra came up. Any help in this direction is appreciated.
QUESTION:
Let $m,n\in\mathbb{N}$ and let $A_1,\ldots, A_m\in M_n(\mathbb{R})$ be real, symmetric, indefinite matrices. I'm interested in conditions on $A_1,\ldots,A_m$ which ensures that the set
$$P:=\big\{\sum_{i=1}^{m}\lambda_i A_i:\lambda_i\in\mathbb{R}\big\}$$
contains a positive-definite matrix. I'm aware of the following result due to Hestenes-McShane (1940) which is suffcient but not necessary.
THEOREM (Hestenes-McShane)
Let $m,n\in\mathbb{N}$ and let $A,B_i\in M_n(\mathbb{R})$ be real symmetric matrices, for all $i=1,\ldots,m$. Let us write, for each $i=1,\ldots,m$,
$$Z_{i}:=\{x\in\mathbb{R}^n:\langle B_i x;x \rangle=0\}$$
Let us suppose that
-
$\langle A x;x \rangle>0$, for all $x\in \cap_{i=1}^{m}Z_i$, $x\neq 0$.
-
$B$ is indefinite on $\mathbb{R}^n$, for all non-zero $B\in\operatorname*{span}\{B_i:i=1,\ldots,m\}$.
-
For every non-zero subspace $S\subseteq \mathbb{R}^n$ satisfying $$S\cap\left(\cap_{i=1}^{m}Z_i\right)=\{0\},$$ there exists $B\in\operatorname*{span}\{B_i:i=1,\ldots,m\}$ such that $B$ is positive definite on $S$.
Then, there exists $B\in\operatorname*{span}\{B_i:i=1,\ldots,m\}$ such that $A-B$ is positive definite on $\mathbb{R}^n$.
Unfortunately, in my case, condition 3 is not satisfied. Has this result been improved later?
Best Answer
The following recent paper: "An exact duality theory for semidefinite programming based on sums of squares" by I. Klep, and M. Schweighofer (both are on MO I think) addresses exactly your question: When is there a $\lambda \in \mathbb{R}^m$ such that $\sum_i \lambda_iA_i \succeq 0$.
If you want something simpler, then the following Lemma, cf. L.Lovasz lecture notes, Lemma 3.2, might be of help (notice $\succ$ instead of $\succeq$).
Without the strict $\succ$ relation, the situation gets trickier (we don't have a perfect Farkas Lemma for SDPs).