Principal skewness analysis (PSA) has been introduced for feature extraction in hyperspectral imagery. As a thirdorder generalization of principal component analysis (PCA), its solution of searching for the local maximum skewness direction is transformed into the problem of calculating the eigenpairs (the eigenvalues and the corresponding eigenvectors) of a coskewness tensor. By combining a fixed-point method with an orthogonal constraint, the new eigenpairs are prevented from converging to the same previously determined maxima. However, in general, the eigenvectors of the supersymmetric tensor are not inherently orthogonal, which implies that the results obtained by the search strategy used in PSA may unavoidably deviate from the actual eigenpairs. In this paper, we propose a new nonorthogonal search strategy to so lve this problem and the new algorithm is named nonorthogonal principal skewness analysis (NPSA). The contribution of NPSA lies in the finding that the search space of the eigenvector to be determined can be enlarged by using the orthogonal complement of the Kronecker product of the previous eigenvector with itself, instead of its orthogonal complement space. We also give a detailed theoretical proof on why we can obtain the more accurate eigenpairs through the new search strategy by comparison with PSA. In addition, after some algebraic derivations, the complexity of the presented algorithm is also greatly reduced. Experiments with both simulated data and real multi/hyperspectral imagery demonstrate its validity in feature extraction.