Spectral clustering is a well-known method for grouping objects by analyzing the spectral properties of a similarity matrix. It is based on graph theory and is particularly suitable for nonlinearly separable cluster structures. Typically, clusters are obtained through a hard partitioning of the data. However, fuzzy assignments allow each data point to belong to more than one cluster with varying degrees of membership, helping the user better understand overlapping structures and the relationships between clusters, especially in cases where clear boundaries do not exist. In this paper, we present a fuzzy spectral clustering algorithm that performs data embedding and clustering simultaneously, rather than using a tandem approach, to improve the quality of the resulting data partition. The proposed method has been applied to two artificial data sets, one with nonlinearly separable clusters and the other with Gaussian clusters.