- #1
Hamza Abbasi
- 47
- 4
While reading problems in my physics book , I encountered a statement very often "Eigen Value Problem" , I read about it from many sources , but wasn't able to understand it . So what exactly is an Eigen Value Problem?
Strilanc said:
whit3r0se- said:Let A=Any vector, x=eigen vector, e=eigen value
The definition of eigenvalue is the following
Ax=ex [where x=eigen vector corresponding to this value],this allows us to find particular values e whereby we can map the vector A into a multiple of itself.
An eigenvalue problem is a mathematical problem that involves finding the values (eigenvalues) and corresponding vectors (eigenvectors) that satisfy a given equation. This equation is typically represented as a matrix equation and is used to study the properties of linear transformations.
Eigenvalue problems have a wide range of applications in various fields such as physics, engineering, and computer science. They are used to solve differential equations, analyze vibration modes in mechanical systems, and in principal component analysis for data analysis and reduction.
Eigenvalue problems are typically solved using matrix algebra and linear algebra techniques. The most common method is the power iteration method, which involves repeatedly multiplying a starting vector by the given matrix until it converges to an eigenvector. Other methods include the QR algorithm and Jacobi method.
Eigenvalues and eigenvectors are closely related in an eigenvalue problem. The eigenvectors represent the directions in which the transformation represented by the matrix stretches or compresses, while the eigenvalues represent the amount of stretching or compressing in those directions.
Yes, eigenvalue problems can have multiple solutions. In fact, the number of eigenvalues and eigenvectors is equal to the dimension of the matrix. This means that a square matrix of size n will have n eigenvalues and n corresponding eigenvectors. Additionally, multiple eigenvectors can have the same eigenvalue.