Eigen Vector and Eigen Value

Abstract

The word Eigen is understood in the German language with the meaning of own, proper, character. But in this discussion, instead of the above words, we use specific or Appropriation words. According to the above explanation, the following concepts are discussed. A definite vector, a definite value (a definite coefficient), a definite space, a definite equation, a definite polynomial, a definite matrix, and a definite function. The concept of Eigenvector and Eigenvalue has an important and special rule in linear algebra and functional analysis, which includes spectral theory.

Its eigenvectors are those vectors that are only stretched, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or squished. If the eigenvalue is negative, the eigenvector's direction is reversed.

 The eigenvectors and eigenvalues of a linear transformation serve to characterize it, and so they play important roles in all the areas where linear algebra is applied, from geology to quantum mechanics. In particular, it is often the case that a system is represented by a linear transformation whose outputs are fed as inputs to the same transformation (feedback). In such an application, the largest eigenvalue is of particular importance, because it governs the long-term behavior of the system after many applications of the linear transformation, and the associated eigenvector is the steady state of the system.