eigenvalues and eigenvectors

by Pseudo Epsilon
Tags: eigenvalues, eigenvectors
 P: 102 can someone PLEASE explain eigenvalues and eigenvectors and how to calculate them or a link to a site that teaches it simply?
 P: 102 Ive already read the wiki and asked my math teacher, he doesnt even know what they are.
 P: 20 I think that Khan Academy does a great job explaining just that! http://www.khanacademy.org/math/line...d-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.
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eigenvalues and eigenvectors

Do you know what "vectors" and "linear transformations" are? Do you know what a "linear vector space" is?
P: 266
 Quote by Pseudo Epsilon Ive already read the wiki and asked my math teacher, he doesnt 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.
 P: 102 dont judge me but how does one map one vector space onto another?
 P: 102 he doesnt know what a vector space even is! And the wiki doesnt do much to even seperate it from vectors.
 PF Patron Sci Advisor Thanks P: 3,975 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##.
 P: 102 thanks!

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