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Positive Definite Matrix

  1. Jun 30, 2016 #1
    1. The problem statement, all variables and given/known data
    a positive definite matrix has eigenvalues λ=1 and λ=2. find the matrix

    2. Relevant equations


    3. The attempt at a solution
    I've used a 2x2 matrix with entries a0,a1,a2,a3 as the unknown matrix but no use. (As little as i know a0 and a3 should be 1 and 2 respectively; corresponding to the eigenvalues) Any ideas on how to solve this?
     
    Last edited: Jun 30, 2016
  2. jcsd
  3. Jun 30, 2016 #2

    fresh_42

    Staff: Mentor

    The fact, that the matrix is ##2\times 2## should be in the list of conditions I guess.
    Start at the beginning: What does it mean for ##\begin{pmatrix}a && b \\ c && d \end{pmatrix}## to be positive definite?
    (I find this notation easier in this case for I'll have to type less indices.)
    Can you really assume already ##a=1## and ##d = 2##? What do you really know about ##λ_1 \cdot λ_2 = 2## and ##λ_1 + λ_2=3## with respect to the characteristic polynomial?
     
  4. Jun 30, 2016 #3
    I've assumed it to be a 2x2 matrix as a positive 2x2 matrix would have only two eigenvalues. No other conditions are provided for this problem......
     
  5. Jun 30, 2016 #4

    fresh_42

    Staff: Mentor

    Yes, but you have not mentioned whether ##1## and ##2## are the only eigenvalues nor their multiplicity. Anyway, can you answer my questions?
     
  6. Jun 30, 2016 #5

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    The concepts of "definiteness" (positive, negative, semi-, etc) apply to symmetric matrices, so your matrix has three unknown parameters in it:
    [tex] A = \pmatrix{a & b \\b & c} [/tex]
    Its characteristic polynomial is a quadratic function of ##\lambda## having roots 1 and 2. What does that tell you about ##a,b,c##?

    Why would you think that your ##a_0## and ##a_3## should be 1 and 2 (or 2 and 1)?
     
  7. Jun 30, 2016 #6

    fresh_42

    Staff: Mentor

    This is not necessary to be positive definite. The definition applies to all bilinear forms and every quadratic matrix defines one.
    (I'm not quite sure whether I have made some mistake or not, but my solution isn't symmetric. The elements on the second diagonal may cancel out.)
     
    Last edited: Jun 30, 2016
  8. Jun 30, 2016 #7
    The product of eigenvalues is equal to the determinant of a matrix, and the sum of eigenvalues is equal to the trace. and the determinant of a 2 × 2 matrix A − λI can be written as:
    λ^2 − (trace)λ + determinant = 0
     
  9. Jun 30, 2016 #8

    fresh_42

    Staff: Mentor

    Correct. So the characteristic polynomial has to be ##char(t) = t^2-3t+2 = \det\begin{pmatrix}a-t && b \\ c && d-t\end{pmatrix}.##
    (I've changed the variable to ##t## because you already used ##λ## for the eigenvalues, i.e. the solutions of ##char(t)=0.##)
    Furthermore the definition of positive definiteness means, that ##(x,y)\begin{pmatrix}a && b \\ c && d\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix} > 0## for all possible values of ##x,y## where not both are ##0.##
    These conditions, if written out, can be used to calculate possible values of ##a,b,c,d##. And there is an obvious solution to it and another one.
     
  10. Jun 30, 2016 #9

    Ray Vickson

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    Science Advisor
    Homework Helper

    Whether or not the 'definiteness' concept applies generally, or only to symmetric matrices, is a point of disagreement in the literature.

    Anyway, one can reduce the general form for matrix ##B## to the equivalent symmetric form ##A = (1/2) B + (1/2) B^T##, get the general symmetric solution, then deal with the multitude of solutions to ##b_{12} + b_{21} = 2 a_{12}##. There are still infinitely many symmetric solutions, but all satisfying some specific relationships between ##a_{11} , a_{12} = a_{21}## and ##a_{22}##..
     
  11. Jun 30, 2016 #10
    Thank you all for helping me out :smile:
     
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