Eigenvalues

Under the effect of A transformation, the vector ξ only changes in scale to λ times of the original. ξ is called an eigenvector of A, and λ is the corresponding eigenvalue (eigenvalue), which is a quantity that can be measured (in experiments). Correspondingly, in the theory of quantum mechanics, many quantities cannot be measured. Of course, this phenomenon also exists in other theoretical fields.

Let A be an n-order matrix. If there exists a constant λ and a non-zero n-dimensional vector x such that Ax=λx, then λ is called the eigenvalue of the matrix A, and x is the eigenvector of A belonging to the eigenvalue λ.

Feature vector

Mathematically, an eigenvector of a linear transformation is a non-degenerate vector whose direction is invariant under the transformation. The scale to which the vector is scaled under the transformation is called its eigenvalue. A linear transformation can usually be fully described by its eigenvalues ​​and eigenvectors. An eigenspace is the set of eigenvectors with the same eigenvalue. The word "characteristic" comes from the German word eigen. Hilbert first used the word in this sense in 1904, and Helmholtz used it earlier in a related sense. The word eigen can be translated as "own", "specific to", "characteristic", or "individual". This shows how important eigenvalues ​​are in defining a particular linear transformation.