Eigenvalue Problem with the Basis Exchange Algorithm

The eigenvalue problem plays an important role in contemporary methods of exploratory data analysis. As an example, the principal component analysis (PCA) widely used in data exploration, is based on finding the eigenvalues and eigenvectors of the covariance matrix.

The paper presents a new method of the eigenvalue problem solution which uses the basis exchange algorithms. The basis exchange algorithms, similarly to the linear programming techniques are based on the Gauss-Jordan transformation of the inverted matrices. The proposed approach to the eigenvalue problem may also be connected to the regularization of feature vectors which constitute squared matrices by single unit vectors. The proposed approach is based on inducing a linear dependence among regularized vectors.