Interference alignment(IA)with symbol extensions in the quasi-static flat-fading K-user multipleinput multiple-output(MIMO)interference channel(IC)is considered in this paper.In general,long symbol extensions are required to achieve the optimal fractional degrees of freedom(DOF).However,long symbol extensions over orthogonal dimensions produce structured(diagonal or block diagonal)channel matrices from transmitters to receivers.Most of existing approaches are limited in cases where the channels have some special structures,because they align the interference without preserving the dimensionality of the desired signal explicitly.To overcome this common drawback of most existing IA algorithms,two novel iterative algorithms for IA with symbol extensions are proposed.The first algorithm designs transceivers for IA based on the mean square error(MSE)criterion which minimizes the total MSE of the system while preserving the dimensionality of the desired signal.The novel IA algorithm is a constrained optimization problem which can be solved by Lagrangian method.Its convergence is proven as well.Utilizing the reciprocity of alignment,the second algorithm is proposed based on the maximization of the multidimensional case of the generalized Rayleigh Quotient.It maximizes each receiver’s signal to interference plus noise ratio(SINR)while preserving the dimensionality of the desired signal.In simulation results,we show the superiority of the proposed algorithms in terms of four aspects,i.e.,average sum rate,the fraction of the interfering signal power in the desired signal subspace,bit error rate(BER)and the relative power of the weakest desired data stream. Interference alignment (IA) with symbol extensions in the quasi-static flat-fading K-user multipleinput multiple-output (MIMO) interference channel (IC) is considered in this paper. General, long symbol extensions are required to achieve the optimal fractional degrees of freedom (DOF) .However, long symbol extensions over orthogonal dimensions produce structured (diagonal or block diagonal) channel matrices from transmitters to receivers .Most of existing approaches are limited in cases where the channels have some special structures, because they align the interference without preserving the dimensionality of the desired signal explicitly.To overcome this common drawback of most existing IA algorithms, two novel iterative algorithms for IA with symbol extensions are proposed. The first algorithm designs transceivers for IA based on the mean square error (MSE) criterion which minimizes the total MSE of the system while preserving the dimensionality of the desired signal. The novel IA algorithm is a con strained optimization problem which can be solved by Lagrangian method. Its convergence is proven as well. Utilizing the reciprocity of alignment, the second algorithm is proposed based on the maximization of the multidimensional case of the generalized Rayleigh Quotient. It maximizes each receiver’s signal to interference plus noise ratio (SINR) while preserving the dimensionality of the desired signal. In simulation results, we show the superiority of the proposed algorithms in terms of four aspects, ie, average sum rate, the fraction of the interfering signal power in the desired signal subspace, bit error rate (BER) and the relative power of the weakest desired data stream.