Solution of eigenvector: Starting from the definition, Ax=cx, A is matrix, C is eigenvalue, and X is eigenvector.

The eigenvector of matrix is actually one of the main viewpoints of matrix, which is widely used. Mathematically, the characteristic vector of linear transformation is a nondegenerate vector, and its target is stable under this transformation. The scaling ratio of this vector under this transformation is called its eigenvalue.

The eigenvector of linear transformation refers to a non-zero vector that is stable for transformation subscripts, or is easily multiplied by a scale factor. The eigenvalues corresponding to the eigenvectors are the scale factors multiplied by them. Feature space is a space composed of all feature vectors with similar eigenvalues, and it also contains zero vector, but it should be noted that zero vector itself is not a feature vector.

The principal eigenvector of linear transformation is the eigenvector corresponding to the largest eigenvalue, the geometric multiplicity of the eigenvalue is the dimension of the corresponding eigenvalue space, and the spectrum of linear transformation in the finite-dimensional vector space is the set of all its eigenvalues.

Application of eigenvectors;

1, factor analysis

In factor analysis, the eigenvector of covariant matrix corresponds to the factor, and the eigenvalue is the factor load. Factor analysis is a statistical technique, which is used in social science, market analysis, product management, operation planning and so on. Its purpose is to explain the changes of some observable random variables with a few unobservable random variables called elements.

2. Characteristic face

In image processing, face image processing can be regarded as a vector, the weight of this vector is the gray level of each pixel, and the dimension of this vector space is the number of pixels. The eigenvector of the large covariance matrix of standardized face graphics is called eigenface.