Understanding Eigenvalues and Eigenvectors: A Beginner's Guide

[Pseudo Epsilon]

can someone PLEASE explain eigenvalues and eigenvectors and how to calculate them or a link to a site that teaches it simply?

 

[Pseudo Epsilon]

Ive already read the wiki and asked my math teacher, he doesn't even know what they are.

 

[UVW]

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.

 

[HallsofIvy]

Do you know what "vectors" and "linear transformations" are? Do you know what a "linear vector space" is?

 

[HomogenousCow]

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.

 

[Pseudo Epsilon]

dont judge me but how does one map one vector space onto another?

 

[Pseudo Epsilon]

he doesn't know what a vector space even is! And the wiki doesn't do much to even separate it from vectors.

 

[WannabeNewton]

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 $\lambda = \pm 1$.

 

[Pseudo Epsilon]

thanks!