  We consider the $K$-user Multiple Input Multiple Output (MIMO) Gaussian
interference channel with $M$ antennas at each transmitter and $N$ antennas at
each receiver. It is assumed that channel coefficients are constant and are
available at all transmitters and at all receivers. The main objective of this
paper is to characterize the Degrees of Freedom (DoF) for this channel. Using a
new interference alignment technique which has been recently introduced in
\cite{abolfazl-final}, we show that $\frac{MN}{M+N} K$ degrees of freedom can
be achieved for almost all channel realizations. Also, a new upper-bound on the
DoF of this channel is provided. This upper-bound coincides with our achievable
DoF for $K\geq K_u\define\frac{M+N}{\gcd(M,N)}$, where $\gcd(M,N)$ denotes the
greatest common divisor of $M$ and $N$. This gives an exact characterization of
DoF for $M\times N$ MIMO Gaussian interference channel in the case of $K\geq
K_u$.
