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Eigen vector proofs

  1. Feb 6, 2012 #1
    1. The problem statement, all variables and given/known data

    Let M be a symmetric matrix. The eigenvalues of M are real and further M can be
    diagonalized using an orthogonal matrix S; that is M can be written as

    M = S^-1*D*S

    where D is a diagonal matrix.
    (a) Prove that the diagonal elements of D are the eigenvalues of M.
    (b) Prove that the real symmetric matrix M is positive defi nite if and only if its eigen-
    values are positive.

    2. Relevant equations

    Mx=λx


    3. The attempt at a solution

    a) So i know that S is the eigen vectors of M and D is the eigen values but I do not know how to prove that.

    b)Does this have something to do with the gradient. I know positive definite means that transpose(x)*M*x must be greater than zero where x is x(1) through x(n) and M is the matrix in question.
     
  2. jcsd
  3. Feb 6, 2012 #2
    S is not the eigenvectors of M, S^{-1} is, the rest is straightforward verification.
     
  4. Feb 6, 2012 #3
    With what equation would I begin that proof?
     
  5. Feb 6, 2012 #4
    first you claim that S^-1 are the eigenvectors, then you prove your claim by definition of eigenvectors
     
  6. Feb 6, 2012 #5

    Deveno

    User Avatar
    Science Advisor

    note that M = S-1DS means that

    S-1D = MS-1.

    express both matrix products above in terms of the column vectors of the matrix on the right in each product. compare your results, what do they say?
     
  7. Feb 7, 2012 #6
    Ok I have that now what about the positive definite aspect?
     
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