I'm doing some research in Control theory, and a stumbled with this problem. Any help is appreciated.


**QUESTION**
Let $P\_1,\dots,P\_m$ be $m$ symmetric positive definite $n\times n$ matrices with $m<n$ and real entries. I'm looking for necessary conditions for the existence of nontrivial real coefficients $\alpha\_1,\dots,\alpha\_m$ (not all $\alpha\_i$ equal to 0) such that
$$
H(\alpha) = \sum\_{i=1}^m \alpha\_iP\_i
$$
is singular (det$H(\alpha)=0$) and $\sum\_{i=1}^m\alpha\_m=0$.


**More details**


Of course there are many ways to come up with some sufficient conditions. For instance, if $m>3$ one can set $\alpha\_0 = 1, \alpha\_2 = s, \alpha\_3 = (-1-s)$ and $\alpha\_i=0, i>3$ for some $s$, and then solve the polynomial $\det(P\_1 + sP\_2 + (-1-s)P\_3) = 0$ for $s$ (then check one of those values for $s$ is real). There are many other similar ways in which one can come up with other values for $\alpha\_1,\dots,\alpha\_m$. But I'm interested for a necessary condition which one can check by looking at the matrices $P\_1,\dots,P\_m$ and see if there exists such $\alpha\_1,\dots,\alpha\_m$. Such conditions are easy to derive when $m\leq 3$ (using a similar reasoning as the example before), but I'm interested in the general case.


Of course I know that this may be hard in general, but any help, references and suggestions are appreciated.



Necessary conditions for existence of linear combination of these matrices to be singular
Eigenvalues are continuous functions of the matrix entries if this is expressed carefully. Consider $H(c) = cP\_1+P\_2+\cdots+P\_m$. When $c$ is large and negative, the eigenvalues of $H(c)$ are all negative. When $c$ is positive, $H(c)$ is psd and the eigenvalues are all positive. So there is some $c$ such that $H(c)$ has a zero eigenvalue.


EDIT. Nathaniel noted that I had missed the condition $\sum\_i \alpha\_i=0$. Oops. I'll try again.


Assume $m=3$ (as we can take $\alpha\_i=0$ for $i\ge 4$). Choose arbitrary non-zero $\beta\_1,\beta\_2,\beta\_3$ summing to 0. Choose continuous functions $f\_1,f\_2,f\_3:[0,1]\to\mathbb R$ such that, for each $i$, $f\_i(0)=-\beta\_i$ and $f\_i(1)=\beta\_i$, and also that $f\_1(t),f\_2(t),f\_3(t)$ always sum to 0 but are never all 0 at the same time. (Linear functions won't do but almost anything else will do.)


Define $H\_t=f\_1(t)P\_1 + f\_2(t)P\_2 + f\_3(t)P\_3$.


First consider matrices of odd order. If $H\_0$ and $H\_1$ are nonsingular, their determinants have opposite sign, so $H\_t$ is singular for some $t$ by continuity.


For matrices of even order, the same is true if the numbers of positive and negative eigenvalues of $H\_0$ are different. But if those numbers are the same I'm not sure what to do. Can we always choose $\beta\_1,\beta\_2,\beta\_3$ so that $H\_0$ has different numbers of positive and negative eigenvalues?


