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Matrix properties theorem

  1. Feb 17, 2012 #1
    1. The problem statement, all variables and given/known data

    Theorem:
    Suppose A, B, C, and D are matrices of the same size. Then
    a) If A ≤ B and B ≤ C, then A ≤ C
    b) If A ≤ B and C ≤ D, then A + C ≤ B + D
    c) If A ≤ B, then cA ≤ cB for any positive constant c and cA >= cB for any negative constant c

    Prove this theorem. Must use arbitrary matrices, one where both the size and entries are specified as variables

    2. Relevant equations



    3. The attempt at a solution

    Let A=[ai,j], B=[bi,j], C=[ci,j], D=[di,j]

    a) Let (1) [ai,j] ≤ [bi,j]
    (2) [bi,j] ≤ [ci,j]
    Adding (1) and (2), we get
    [ai,j] + [bi,j] ≤ [bi,j] + [ci,j]
    Subtracting [bi,j] from both sides,
    [ai,j] ≤ [ci,j]
    Therefore A ≤ C

    b) Let (1) [ai,j] ≤ [bi,j]
    (2) [ci,j] ≤ [di,j]
    Adding (1) and (2), we get
    [ai,j] + [ci,j] ≤ [bi,j] + [di,j]
    Therefore A + C ≤ B + D

    c) Not really sure how to do c?
     
    Last edited by a moderator: Feb 17, 2012
  2. jcsd
  3. Feb 17, 2012 #2
    what is the definition of <= for matrices?
     
  4. Feb 17, 2012 #3
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  5. Feb 17, 2012 #4
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  6. Feb 17, 2012 #5

    vela

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    First, tell us what it means to say matrix A ≤ matrix B. You have to know your definitions in math.
     
  7. Feb 17, 2012 #6
    OK. this proof is in my linear programming class. I cannot remember what this means.. she did not give us a recap on inequalities of matrices
     
    Last edited by a moderator: Feb 17, 2012
  8. Feb 17, 2012 #7
    Ya, you are on the right track and "c" is related to inequalities in algebra.
     
  9. Feb 17, 2012 #8

    vela

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    Then how can you possibly know what properties matrix inequalities have? Like how do you know that A+B ≤ B+C implies A ≤ C? It's true for real numbers, but you can't automatically assume it holds for matrices.

    Don't you have a textbook you can consult for basic definitions?
     
  10. Feb 17, 2012 #9
  11. Feb 17, 2012 #10
    nope my textbook just goes into the matrices of LP problems, not their properties :(
    i just assumed it worked with matrices as it does with regular numbers, since any entry in A or B is just a number... so my attempt is completely wrong?
     
    Last edited by a moderator: Feb 17, 2012
  12. Feb 17, 2012 #11
    Well, it depends. If I had to guess, perhaps A, B, C, D are square matrices of the same size and the ordering is on the determinants of those matrices?

    Could you post more context?
     
  13. Feb 17, 2012 #12

    micromass

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  14. Feb 17, 2012 #13
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    Last edited: Feb 17, 2012
  15. Feb 17, 2012 #14
    In (a), for all i,j, ai,j ≤ bi,j, and bi,j ≤ ci,j implies what about the relationship between the elements of A and C?

    Proceed similarly for the other parts.

    dirk_mec1's link may be helpful. micromass' link delves a bit too deep to be useful.

    If you want, you can go deeper with respect to this problem. The definition is an example of a http://en.wikipedia.org/wiki/Partial_order
     
  16. Feb 17, 2012 #15
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  17. Feb 17, 2012 #16

    vela

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    Notation-wise, I'd remove the square brackets because you're talking about the specific elements of each matrix. Note that they don't use the square brackets in the definition you cited. Otherwise, they look good.
     
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