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.