Graphical representations provide us with a tool allowing visual inspection of the sequences. To visualize and compare different DNA sequences, a novel alignment-free method is proposed in this paper for both graphical representation and similarity analysis of sequences. We introduce a transformation to represent each DNA sequence with neighboring nucleotide matrix. Then, based on approximate joint diagonalization theory, we transform each DNA primary sequence into a corresponding eigenvalue vector (EVV), which can be considered as numerical characterization of DNA sequence. Meanwhile, we get graphical representation for DNA sequence via the plot of EVV in 2-D plane. Moreover, using k-means, we cluster these feature curves of sequences into several reasonable subclasses. In addition, similarity analyses are performed by computing the distances among the obtained vectors. This approach contains more sequence information, and it analyzes all the involved sequence information jointly rather than separately. A typical dendrogram constructed by this method demonstrates the effectiveness of our approach.