Eigenvalue Problem: What Is It?

In summary, the conversation discussed the concept of eigenvalue problems in physics and how they are related to linear operators and differential equations. The conversation also mentioned the importance of understanding linear algebra in order to better comprehend eigenvalues and eigenvectors. The question of what exactly an eigenvalue problem is was also raised.
  • #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?
 
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  • #3
  • #4
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.
 
  • #5
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.

## A ## is not a vector. It is a linear operator.

The eigenvalue problem is: given a linear operator ## A ## (in a matrix form or otherwise), find it eigenvalues ## \lambda ## and eigenvectors ## u ## defined as $$ Au = \lambda u. $$ Intuitively, an eigenvector is a vector that does not change its direction (except when ## \lambda ## is negative it got flipped) upon the action of ## A ##. Examples of eigenvalue problems ubiquitous in physics are finding normal modes of wave equations or stationary states of the Schrödinger equation in quantum mechanics. In these examples, ## \lambda ## are frequencies and energies respectively. They are eigenvalue problems because differential operators, ##\frac{d}{dx}## and linear combinations of its powers, are linear operators: the derivative of the sum is the same as the sum of derivatives.
 
  • #6
In order to get a feel for what eigenvalues and eigenvectors are, it is very instructive to look at linear transformations in two dimensions. Are you familiar with the basics of linear algebra (vectors, matrices, changing bases, etc.)?
 
  • #7
I think the OP was asking what the problem is not what Eigenvectors and values are. I think the problem is that it is problem not problem. Damn english!
 

What is an eigenvalue problem?

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.

What is the importance of eigenvalue problems?

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.

How are eigenvalue problems solved?

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.

What is the relationship between eigenvalues and eigenvectors?

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.

Can eigenvalue problems have multiple solutions?

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.

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