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


1. $\langle A x;x \rangle>0$, for all $x\in \cap\_{i=1}^{m}Z\_i$, $x\neq 0$.
2. $B$ is indefinite on $\mathbb{R}^n$, for all non-zero $B\in\operatorname\*{span}\{B\_i:i=1,\ldots,m\}$.
3. 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?



On the positive definiteness of a linear combination of matrices
The following recent paper: ["An exact duality theory for semidefinite programming based on sums of squares"](http://www.math.uni-konstanz.de/~schweigh/publications/sosdualsdp.pdf) 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](http://www.cs.elte.hu/~lovasz/semidef.ps), might be of help (notice $\succ$ instead of $\succeq$).



> 
> **Lemma.** Let $A\_i$ be real symmetric matrices. Then, the set $P\_+ := \lbrace\sum\_i \lambda\_i A\_i \succ 0\rbrace$ is empty if and only if there exists a semidefinite matrix $X \neq 0$, such that $\mbox{trace}(A\_iX) = 0$ for all $i$.
> 
> 
> 


Without the strict $\succ$ relation, the situation gets trickier (we don't have a perfect Farkas Lemma for SDPs).


