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Homework Help: Eigenvalue questions

  1. Jul 7, 2011 #1
    1. The problem statement, all variables and given/known data
    Consider the matrix
    A=[a d f; 0 b e; 0 0 c], where all elements are real numbers
    (a) what condition(s) on the elements of A are sufficient to guarantee that A has 3 distinct eigenvalues?
    (b) prove that any two eigenvectors x1 and x2 associated with two distinct eigenvalues e1=e2 must be linearly independent
    (c) what condition(s) on the elements of A are sufficient to guarantee that the inverse A^-1 exists?
    (d) consider diff eq
    d/dt(u)=Au, u(0)=u0
    where A is the matrix discussed above with three distinct eigenvalues, and u is a vector. write the general solution u(t) in terms of the eigenvalues and eigenvectors of A. do not solve for the actual eigen vectors.
    (e) prove that a soln u(t) that is initially parallel to an eigenvector must remain so for all time.


    2. Relevant equations
    will involve diagonal matrices
    If for a given matrix there exists a matrix B such that AB=I, then B=A^-1, if I is the identity matrix.

    3. The attempt at a solution
    (a) It seems that in order for A to have three distinct eigenvalues, a and b and c cannot be equal to each other- I think if that diagonal relationship is satisfied, the values will be distinct. Not sure though.
    (b) Not sure how to approach this.
    (c) I think I am supposed to use the relevant equation 2 that I wrote to prove this- would this be satisfied at all times if the matrix is a diagonal matrix? In that case, d, f, and e should be zero?
    (d) Not sure
    (e) Not sure
     
  2. jcsd
  3. Jul 7, 2011 #2

    vela

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    This is correct. Can you prove it? Try calculating what the eigenvalues of A are.
    One way you can do it is assume x1 and x2 are linearly dependent and show it leads to a contradiction.
    This isn't correct. Hint: What can you say about the determinant of an invertible matrix?
    You should review how to solve systems of linear differential equations.
     
  4. Jul 7, 2011 #3

    epenguin

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    The equation for the eigenvalues det(A - λI) = 0 very easily gives you the condition you conjecture.
     
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