Understanding Eigenvalues and Eigenvectors: A Beginner's Guide

In summary, eigenvalues and eigenvectors are important concepts in linear algebra that involve finding special vectors and values for a given linear operator. They can be easily explained and understood through resources such as Khan Academy, and it is recommended to seek help from forums and other sources if a teacher is unable to provide proper guidance.
  • #1
Pseudo Epsilon
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0
can someone PLEASE explain eigenvalues and eigenvectors and how to calculate them or a link to a site that teaches it simply?
 
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  • #2
Ive already read the wiki and asked my math teacher, he doesn't even know what they are.
 
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  • #3
I think that Khan Academy does a great job explaining just that!

http://www.khanacademy.org/math/linear-algebra/alternate_bases/eigen_everything/v/linear-algebra--introduction-to-eigenvalues-and-eigenvectors

Also, don't forget that there's a "Math & Science Learning Materials" forum on this website; it might be a better place to check in the future.
 
  • #4
Do you know what "vectors" and "linear transformations" are? Do you know what a "linear vector space" is?
 
  • #5
Pseudo Epsilon said:
Ive already read the wiki and asked my math teacher, he doesn't even know what they are.
That is sad to hear, eigenvectors and eigenvalues are very basic maths. Teachers are very underqualified these days.

A linear operator is a function that maps one vector space into another, there are certain vectors which when transformed by the linear operator, comes out as a scalar multiple of itself, the vector is the eigenvector and the multiple is the eigenvalue.
 
  • #6
dont judge me but how does one map one vector space onto another?
 
  • #7
he doesn't know what a vector space even is! And the wiki doesn't do much to even separate it from vectors.
 
  • #8
Let ##V## be a vector space over ##F## and let ##T:V\rightarrow V## be a linear operator. We say ##v\in V\setminus \left \{ 0 \right \}## is an eigenvector of ##T## if there exists a ##\lambda\in F## such that ##T(v) = \lambda v##. We call ##\lambda## an eigenvalue of ##T##.

As an example, let ##V = M_{n\times n}(\mathbb{R})## and let ##T:V\rightarrow V,A \mapsto A^{T}##. We want to find the eigenvalues of ##T##. Let ##A\in V## such that ##T(A) = A^{T} = \lambda A##. Note that ##T(T(A)) = \lambda ^{2}A = (A^T)^T = A## hence ##A(\lambda^{2} - 1) = 0## and since eigenvectors have to be non-zero, this implies ##\lambda = \pm 1##.
 
  • #9
thanks!
 

1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are concepts in linear algebra that are used to describe certain properties of a square matrix. Eigenvalues are scalar values that represent the amount by which an eigenvector is scaled when it is multiplied by the matrix. Eigenvectors are non-zero vectors that remain in the same direction when multiplied by the matrix.

2. How are eigenvalues and eigenvectors calculated?

Eigenvalues and eigenvectors are calculated by finding the roots of the characteristic polynomial of the matrix. The characteristic polynomial is obtained by subtracting the identity matrix multiplied by a scalar from the original matrix and then finding the determinant of the resulting matrix.

3. What is the significance of eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are important because they provide insight into the behavior of a matrix. They are used to solve systems of linear equations, analyze systems of differential equations, and perform transformations in linear algebra.

4. Can a matrix have more than one eigenvalue?

Yes, a matrix can have multiple eigenvalues. However, each eigenvalue will correspond to a unique eigenvector. This means that the number of eigenvalues a matrix has is equal to the number of eigenvectors.

5. How are eigenvalues and eigenvectors used in data analysis?

Eigenvalues and eigenvectors are commonly used in data analysis for dimensionality reduction and feature extraction. They can also be used to identify patterns and relationships between variables in a dataset.

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