  Let $A(x)=A\_0+x\_1A\_1+...+x\_nA\_n$ be a linear matrix, or pencil,
generated by given symmetric matrices $A\_0,A\_1,...,A\_n$ of size $m$ with
rational entries. The set of real vectors x such that the pencil is positive
semidefinite is a convex semi-algebraic set called spectrahedron, described by
a linear matrix inequality (LMI). We design an exact algorithm that, up to
genericity assumptions on the input matrices, computes an exact algebraic
representation of at least one point in the spectrahedron, or decides that it
is empty. The algorithm does not assume the existence of an interior point, and
the computed point minimizes the rank of the pencil on the spectrahedron. The
degree $d$ of the algebraic representation of the point coincides
experimentally with the algebraic degree of a generic semidefinite program
associated to the pencil. We provide explicit bounds for the complexity of our
algorithm, proving that the maximum number of arithmetic operations that are
performed is essentially quadratic in a multilinear B\'ezout bound of $d$. When
$m$ (resp. $n$) is fixed, such a bound, and hence the complexity, is polynomial
in $n$ (resp. $m$). We conclude by providing results of experiments showing
practical improvements with respect to state-of-the-art computer algebra
algorithms.
