Eigenvalues and eigenvectors

  1. can someone PLEASE explain eigenvalues and eigenvectors and how to calculate them or a link to a site that teaches it simply?
     
  2. jcsd
  3. Ive already read the wiki and asked my math teacher, he doesnt even know what they are.
     
    Last edited by a moderator: May 5, 2013
  4. HallsofIvy

    HallsofIvy 40,307
    Staff Emeritus
    Science Advisor

    Do you know what "vectors" and "linear transformations" are? Do you know what a "linear vector space" is?
     
  5. 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 doesnt know what a vector space even is! And the wiki doesnt do much to even seperate it from vectors.
     
  8. WannabeNewton

    WannabeNewton 5,767
    Science Advisor

    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!
     
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