Linear Algebra - Matrix with given eigenvalues

  1. 1. The problem statement, all variables and given/known data
    Come up with a 2 x 2 matrix with 2 and 1 as the eigenvalues. All the entries must be positive.
    Then, find a 3 x 3 matrix with 1, 2, 3 as eigenvalues.

    3. The attempt at a solution
    I found the characteristic equation for the 2x2 would be λ2 - 3λ + 2 = 0. But then I couldn't get a matrix with positive entries to work for that.
     
    Last edited: Apr 4, 2012
  2. jcsd
  3. Dick

    Dick 25,822
    Science Advisor
    Homework Helper

    Pick a diagonal matrix.
     
  4. Does that count for the entries being positive though?
     
  5. Dick

    Dick 25,822
    Science Advisor
    Homework Helper

    Not really, no. Sorry. Better give this more thought than I gave this response.
     
  6. thanks though!
     
  7. micromass

    micromass 18,695
    Staff Emeritus
    Science Advisor
    Education Advisor

    The 2x2-case is not so difficult. Remember (or prove) that the characteristic polynomail of a 2x2-matrix A is

    [tex]\lambda^2-tr(A)\lambda+det(A)[/tex]

    By the way, I think your characteristic polynomial is wrong.
     
  8. HallsofIvy

    HallsofIvy 40,386
    Staff Emeritus
    Science Advisor

    ??? Why do the diagonal matrices
    [tex]\begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}[/tex]
    and
    [tex]\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\end{bmatrix}[/tex]
    NOT count as "all entries postive"?
     
  9. micromass

    micromass 18,695
    Staff Emeritus
    Science Advisor
    Education Advisor

    He probably doesn't consider 0 to be positive.
     
  10. HallsofIvy

    HallsofIvy 40,386
    Staff Emeritus
    Science Advisor

    But it is much easier to claim that 0 is positive!:tongue:

    Thanks.
     
  11. Oh, I had typed 3 instead of 2 for the characteristic polynomial. I ended up looking at this from a Hermitian matrix point of view.
    And then I got the matrix:
    0 i +1
    i-1 3
    And I did get the right eigenvalues from that. Does that work?
     
  12. micromass

    micromass 18,695
    Staff Emeritus
    Science Advisor
    Education Advisor

    You still have 0 as an entry, you don't want that.
     
  13. Yeah, I didn't realize that at first. :/
     
  14. Dick

    Dick 25,822
    Science Advisor
    Homework Helper

    I don't think i+1 would be considered a positive number either. Stick to real entries. Your diagonal entries need to sum to 3, and their product should be greater than 2. Do you see why?
     
  15. Yes. Theoretically, I know what it should do. I just can't actually find the right values to do it.
     
  16. Dick

    Dick 25,822
    Science Advisor
    Homework Helper

    Call one diagonal entry x. Then the other one must be 3-x. Can you find a positive value of x that makes x*(3-x)>2? Graph it.
     
  17. Well, any value of x between 1 and 2 (like 1.1) work.
     
  18. Dick

    Dick 25,822
    Science Advisor
    Homework Helper

    Ok, so you just need to fill in the rest of the matrix.
     
  19. but if I set x to be 1.1, my matrix would be
    1.1 __
    __ 1.9

    And those two spaces have to be equivalent to 1.1*1.9 - 2, right?
    because no matter what values I try, when the eigenvalues are getting closer to 1 and two, the matrix is just getting closer to the matrix of:
    1 0
    0 2
     
  20. Dick

    Dick 25,822
    Science Advisor
    Homework Helper

    The two spaces multiplied together have to give you 1.1*1.9 - 2. How about putting one blank to be 1 and the other to be 1.1*1.9 - 2? The eigenvalues should work out to EXACTLY 1 and 2. Try it with x=3/2.
     
  21. I had just figured out that
    1.5 0.5
    0.5 1.5
    worked out! :)
     
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook

Have something to add?