  We study the exactness of the semidefinite programming (SDP) relaxation of
quadratically constrained quadratic programs (QCQPs). With the aggregate
sparsity matrix from the data matrices of a QCQP with $n$ variables, the rank
and positive semidefiniteness of the matrix are examined. We prove that if the
rank of the aggregate sparsity matrix is not less than $n-1$ and the matrix
remains positive semidefinite after replacing some off-diagonal nonzero
elements with zeros, then the standard SDP relaxation provides an exact optimal
solution for the QCQP under feasibility assumptions. In particular, we
demonstrate that QCQPs with forest-structured aggregate sparsity matrix, such
as the tridiagonal or arrow-type matrix, satisfy the exactness condition on the
rank. The exactness is attained by considering the feasibility of the dual SDP
relaxation, the strong duality of SDPs, and a sequence of QCQPs with perturbed
objective functions, under the assumption that the feasible region is compact.
We generalize our result for a wider class of QCQPs by applying simultaneous
tridiagonalization on the data matrices. Moreover, simultaneous
tridiagonalization is applied to a matrix pencil so that QCQPs with two
constraints can be solved exactly by the SDP relaxation.
