  In this paper, we study the Helmholtz transmission eigenvalue problem for
inhomogeneous anisotropic media with the index of refraction $n(x)\equiv 1$ in
two and three dimension. Starting with a nonlinear fourth order formulation
established by Cakoni, Colton and Haddar [2009], by introducing some auxiliary
variables, we present an equivalent mixed formulation for this problem,
followed up with the finite element discretization. Using the proposed scheme,
we rigorously show that the optimal convergence rate for the transmission
eigenvalues both on convex and nonconvex domains can be expected. Moreover, by
this scheme, we will obtain a sparse generalized eigenvalue problem whose size
is so demanding even with a coarse mesh that its smallest few real eigenvalues
fail to be solved by the shift and invert method. We partially overcome this
critical issue by deflating the almost all of the $\infty$ eigenvalue of huge
multiplicity, resulting in a drastic reduction of the matrix size without
deteriorating the sparsity. Extensive numerical examples are reported to
demonstrate the effectiveness and efficiency of the proposed scheme.
