How to find the eigenvector

Usually, finding eigenvalues and eigenvectors is to find which vectors (eigenvectors, of course) this matrix can only stretch and to what extent (eigenvalue size). The significance of this is to see clearly which aspects of a matrix can produce the greatest effect, and to discuss and study according to each generated eigenvector (generally those with the largest eigenvalue).

How to find the eigenvector;

Once the eigenvalue λ is found, the corresponding eigenvector can be obtained by solving the eigenvalue equation (a–λ I) v = 0, where V is the eigenvector to be found and I is the identity matrix.

An example of a matrix without real eigenvalues is to rotate 90 degrees clockwise.

Numerical calculation:

In practice, the eigenvalues of large matrices cannot be calculated by characteristic polynomials, and it takes a lot of resources to calculate polynomials. For high-order polynomials, it is difficult to calculate and express the exact "symbolic" roots. Abel-Ruffini theorem shows that the roots of high-order (5 or higher) polynomials cannot be simply expressed by square roots of order N. There are effective algorithms to estimate the roots of polynomials, but small errors of eigenvalues will lead to large errors of eigenvectors. The general algorithm for finding the zero point of characteristic polynomial, that is, eigenvalue, is iterative method. The simplest method is the power method: take a random vector v and then calculate a series of unit vectors.

This sequence almost always converges to the eigenvector corresponding to the eigenvalue with the largest absolute value. This algorithm is simple, but it is not very useful. But QR algorithm and other algorithms are based on this.

Introduction of eigenvectors

The eigenvector is a nondegenerate vector, and its direction remains unchanged under this transformation. The scaling ratio of a vector under this transformation is called its eigenvalue. Eigenvalue is an important concept in linear algebra.

Linear transformation can usually be completely described by its eigenvalues and eigenvectors. Feature space is a set of feature vectors with the same eigenvalue. The word "feature" comes from the German word eigen.

Hilbert first used this word in 1904, and Helm Holtz used it in a related sense earlier. Eigenvalues can be translated into "own", "specific", "characteristic" or "individual", which shows the importance of eigenvalues in defining a specific linear transformation.