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Suppose I have a generic 3-dimensional affine linear subspace of the 6-dimensional space of symmetric $3 \times 3$ matrices. Does such a space necessarily intersect the set of definite (either positive or negative definite) symmetric matrices?

Genericity is important here; this is certainly not necessarily true for a linear subspace, or for spaces parallel to coordinate planes, for example. But will a small perturbation of the affine subspace always suffice to obtain a nontrivial intersection?

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No. Consider the 3-dimensional affine subspace consisting of all matrices having $1,1,-1$ on the diagonal (non-diagonal entries are arbitrary). A small perturbation of this subspace cannot intersect the cone of definite matrices. Indeed, a matrix from a perturbed subspace either has diagonal entries close to $1,1,-1$, or the maximum non-diagonal entry is much larger than the diagonal ones (and in this case, the respective $2\times 2$ minor is negative).

Even 4-dimensional subspace is not enough. Indeed, let $C$ denote the set of nonnegative matrices. It is a sharp (i.e. not containing straight lines) closed convex cone in the 6-dimensional space of all symmetric matrices. Therefore it is a cone over a compact convex set $K$ lying in a 5-dimensional affine subspace (e.g. in the subspace of matrices with trace 1). In this affine subspace, there is a 4-dimensional subspace $L$ which does not intersect $K$ (for example, the set of matrices with trace 1 and the upper-left entry equal to 2). Consider the distance from the set $C\cup(-C)$ as a function on $L$. It is positive and have a linear growth rate at infinity, and these properties are preserved under small perturbations of $L$.

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