What are eigenvalues and eigenvectors?

Click For Summary
Eigenvalues and eigenvectors are fundamental concepts in linear algebra related to matrices and linear transformations. An eigenvector is a non-zero vector that, when a matrix A is applied to it, results in a scalar multiple of itself, expressed as Av = λv, where λ is the eigenvalue. For instance, in a 180-degree rotation transformation in R^2, every vector becomes its negative, making all non-zero vectors eigenvectors with an eigenvalue of -1. Understanding these concepts allows for simplifying complex linear transformations into manageable forms. This simplification is crucial for solving various mathematical and engineering problems.
orochimaru
hi,
i have trouble understanding these two terms.
can anyone explain to me eigenvalues and eigenvectors in laymen terms?

Thks in advance! :smile:
 
Mathematics news on Phys.org
If you have a matrix A (or linear transformation, operator etc.) from the vector space V to itself acting no a vector v, then it will give another vector in the same space.
Generally this vector Ax will be some different vector, one that is linearly independent from v (it points in another direction). However if it is some scalar multiple of v (so Av=\lambda v for some scalar \lambda then v is called an eigenvector (the nullvector is ruled out as an eigenvector by definition) and \lambda is its corresponding eigenvalue.

For example, if you take a vector in the plane R^2 and your linear transformation A is a rotation about the origin over 180 degrees, then every vector v will point in the opposite direction after the transformation, so Av=-v for all v. So every vector (not 0) is an eigenvector of A with eigenvalue -1.
 
You know, I presume, that any linear transformation can be written as a matrix so that applying the transformation to a vector is the same as multiplying the matrix and the vector.

Finding eigenvalues and eigenvectors is essentially finding for what vectors that matrix multiplication acts just like multiplying the vector by a number. It makes it possible to write the linear transformation as a sum of products of numbers,simplifying any problem involving that transformation.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

MHB Solving Matrix A: Characteristic Equation and Eigenvectors
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad What are the Eigenvalues of the Hermitian Inverse \(H^{-1}\)?
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Unravelling Structure of a Symmetric Matrix
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Can one find a matrix that's 'unique' to a collection of eigenvectors?
  • · Replies 33 ·
2
Replies
33
Views
1K
Prove 9 is eigenvalue of ##T^2\iff## 3 or -3 eigenvalue of ##T##.
  • · Replies 5 ·
Replies
5
Views
2K
If |a> is an eigenvector of A, is f(B)|a> an eigenvector of A?
  • · Replies 17 ·
Replies
17
Views
2K
Degeneracy and non degeneracy in quantum mechanics
  • · Replies 2 ·
Replies
2
Views
1K
Can anyone explain to me why eigenvector here is like this
  • · Replies 3 ·
Replies
3
Views
2K
Why Are There Additional Eigenvalues and Eigenvectors of J3?
  • · Replies 17 ·
Replies
17
Views
2K
How to find the eigenvector for a perturbated Hamiltonian?
  • · Replies 13 ·
Replies
13
Views
3K