What is the significance of eigenvalues and eigenvectors in linear algebra, and how are they used in real-world applications?

Question

What is the significance of eigenvalues and eigenvectors in linear algebra, and how are they used in real-world applications?how these are different from each other ?

Answer 1

A full answer to this question is not going to fit here. Eigenvalues/vectors have a wide variety of real-world applications!

From the theoretical side, suppose you have a linear transformation (or in an introductory course, a matrix that represents the transformation) with the same domain and codomain. An eigenvector is an input (domain) vector that "behaves as simply as possible" when you apply the transformation. More precisely, the eigenvector is transformed into a scalar multiple of itself. The particular scalar multiple is the eigenvalue.

If you use a basis for the domain consisting of eigenvectors, the transformation is "as simple as possible" to compute; this leads to the matrix diagonalization material in a typical intro Linear Algebra course.

An interesting real-world application is computer graphics, where you can characterize the geometric effect of a given linear transformation in terms of its eigenvalues. For example, a transformation from R² → R² with eigenvalues 1 and -1 is always a reflection (with the mirror/axis along the line generated by multiples of the eigenvector with eigenvalue 1).