It is, from many points of view, natural to substitute tuples X of symmetric matrices, instead of scalars, into a monic affine linear pencil L.
Ditto for p a symmetric polynomial in freely non-commuting
variables. The inequality L(X) is positive definite is then the free
analog of a linear matrix inequality and the solution set of this
inequality is a free spectrahedron. Likewise, the set of X such that p(X) is
positive definite is a free basic semi-algebraic set. In contrast to the scalar (commuting) case, a
free basic semi-algebraic set which is also bounded is a free
spectrahedron. On the other hand, a projection of a free spectrahedron need
not be a free semi-algebraic set.
Results, which parallel classical rigidity theorems for analytic mappings
in several complex variables, suggest the possibilities for mapping analytically
a free semi-algebraic set bijectively to a free spectrahedron - thus gaining
convexity - are limited.
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