Geometric significance of eigenvalues and eigenvectors in linear algebra and their potential value in data analysis

Eigenvalues and eigenvectors are important concepts in linear algebra, but little research has been done on their geometric significance and other aspects. Therefore, in order to fill the gap in this field, this paper carries out an in-depth study on the geometric significance of eigenvalues and eigenvectors and the important role they play in data analysis. Taking the 2nd order square matrix as an example, we study the trajectory of the new vector after the linear transformation of the unit vector, and on the basis of this, we give the geometric significance of the eigenvalues and eigenvectors of the matrix by taking the invertibility of the matrix as the classification principle. The eigenvalues and eigenvectors and their geometric significance are applied to data analysis examples, and the principal component analysis method and spectral clustering algorithm are constructed and analyzed for practical applications. Under the correction of principal component analysis, the accuracy of recovered spectra of spatial aberration interferometry data is significantly improved, in which the mean square error values of a certain three rows of selected data are reduced by 75.55%, 77.31%, and 77.34%, respectively, compared with those before correction. At the same time, the improved spectral clustering algorithm is able to realize the automatic selection of feature vectors, and the accuracy of the division of the karate club network is also as high as 98.03%. Through the demonstration of practical applications in two different fields, this paper intuitively illustrates the significant role of eigenvalues and eigenvectors and their geometrical significance in data analysis.