What are eigenvalues and eigenvectors?

In summary, eigenvalues and eigenvectors are terms used in linear algebra to describe the behavior of a transformation on a vector in the same vector space. An eigenvector is a vector that, when multiplied by the transformation, results in a scalar multiple of itself, and the corresponding eigenvalue is the scalar multiple. This concept is useful for simplifying problems involving linear transformations.
  • #1
orochimaru
hi,
i have trouble understanding these two terms.
can anyone explain to me eigenvalues and eigenvectors in laymen terms?

Thks in advance! :smile:
 
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  • #2
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 [itex]Av=\lambda v[/itex] for some scalar [itex]\lambda[/itex] then v is called an eigenvector (the nullvector is ruled out as an eigenvector by definition) and [itex]\lambda[/itex] 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.
 
  • #3
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.
 

1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts that are used to describe the behavior of a linear transformation. Eigenvalues are scalar values that represent the amount by which a vector is scaled when transformed by a linear transformation. Eigenvectors are the corresponding vectors that are unchanged in direction when transformed by the linear transformation.

2. How are eigenvalues and eigenvectors useful in science?

Eigenvalues and eigenvectors have many practical applications in science, particularly in physics and engineering. They are used to understand and analyze systems that exhibit linear behavior, such as vibrations, oscillations, and electrical circuits. They also have applications in data analysis, image processing, and machine learning.

3. How do you calculate eigenvalues and eigenvectors?

The process of finding eigenvalues and eigenvectors involves solving a system of linear equations through a technique called diagonalization. This involves finding the characteristic polynomial of a matrix, which is then used to find the eigenvalues. The eigenvectors can then be calculated by solving a system of equations using the eigenvalues.

4. Can a matrix have more than one eigenvalue and eigenvector?

Yes, a matrix can have multiple eigenvalues and corresponding eigenvectors. In fact, the number of eigenvalues and eigenvectors is equal to the dimension of the matrix. However, in some cases, a matrix may have repeated eigenvalues, resulting in fewer distinct eigenvectors.

5. What is the relationship between eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are closely related, as each eigenvector corresponds to a specific eigenvalue. The eigenvalues and eigenvectors of a matrix can provide information about the behavior and properties of the matrix, such as its stability, symmetry, and rank.

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