Kernel based Sparse Representation Classifier (KSRC) can classify images with acceptable performance. In addition, Multiple Kernel Learning based SRC (MKL-SRC) computes the weighted sum of multiple kernels in order to construct a unified kernel while the weight of each kernel is calculated as a fixed value in the training phase. In this paper, an MKL-SRC with non-fixed kernel weights for dictionary atoms is proposed. Kernel weights are embedded as new variables to the main KSRC goal function and the resulted optimization problem is solved to find the sparse coefficients and kernel weights simultaneously. As a result, an atom specific multiple kernel dictionary is computed in the training phase which is used by SRC to classify test images. Also, it is proved that the resulting optimization problem is convex and is solvable via common algorithms. The experimental results demonstrate the effectiveness of the proposed approach.