eigenvalue

under the action of a transformation, the vector ξ only changes in scale to λ times its original size. ξ is said to be an eigenvector of a, λ is the corresponding eigenvalue (eigenvalue), which is a quantity that can be measured (in experiments). correspondingly, in quantum mechanical theory, many quantities cannot be measured. of course , this phenomenon also exists in other theoretical fields.

assume a is an n-order matrix. if there is a constant λ and a non-zero n-dimensional vector x such that ax=λx, then λ is said to be the eigenvalue of matrix a, and x is the eigenvector of a that belongs to the eigenvalue λ.

eigenvector

mathematically, the eigenvector (eigenvector) of a linear transformation is a non-degenerate vector whose direction does not change under the transformation. the proportion by which the vector is scaled under this transformation is called its eigenvalue (eigenvalue). a linear transformation can usually be completely described by its eigenvalues ​​and eigenvectors. eigen space is a collection of eigenvectors with the same eigenvalues. the word "characteristic" comes from the german eigen. hilbert first used the word in this sense in 1904, and earlier helmholtz also used it in a related sense. the word eigen can be translated as "self", "specific to", "characteristic", or "individual". this shows how important eigenvalues ​​are in defining a specific linear transformation.