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Matrix equations

  1. May 19, 2005 #1
    Hi

    I have this following problem:

    Two matrix equations are given

    [tex]C^{T} X = K \ \ Y C^{T} = K[/tex]

    where K, X,Y and C are square matrices. If I wanna calculate X in equation 1 and Y in equation 2 I multiply with [tex]{C^{T}}^{(-1)}[/tex] one both sides of each equation.

    The resulting matrix X in equation is still equal to Matrix Y in equation two ??

    /Fred
     
    Last edited: May 19, 2005
  2. jcsd
  3. May 19, 2005 #2

    arildno

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    Not necessarily!
    Substitute into equation 1 the expression for K from equation 2:
    [tex]C^{T}X=YC^{T}[/tex]
    which, assuming invertibility of [tex]C^{T}[/tex] can be rewritten as:
    [tex]C^{T}X(C^{T})^{(-1)}=Y[/tex]
    Why should we have X=Y???
     
  4. Jun 14, 2005 #3
    Hi but how do I calculate Y in equation 2 ???

    Hope You can help to understand why X could equal Y ?

    Sincerley and Best Regards,

    Fred

    p.s.

    Here are the matrices used in the equations..

    [tex]C = \left[ \begin{array}{ccc} 1 & 1 & 2 \\1 & 2 & 4 \\ 2 & -5 & 2 \end{array} \right][/tex] and [tex]K = \left[ \begin{array}{ccc} 1 & 2 & 4 \\-3 & 2 & 0 \\ -1 & -1 & 2 \end{array} \right][/tex]

     
  5. Jun 14, 2005 #4

    OlderDan

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    Given these two, you can calculate X and Y explicitly and compare them. They are not equal
     
  6. Jun 14, 2005 #5
    Hi and Thank You for Your answer,

    Does [tex]Y = K {C^{T}}^{(-1)}[/tex] ???

     
  7. Jun 14, 2005 #6

    arildno

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    Correct; however, if you haven't got the explicit expression for [tex](C^{T})^{(-1)}[/tex]
    it is better to solve the linear system for the 9 components of Y instead

    (In order for two matrices to be equal, their components must be equal; this gives you 9 equations.)
     
  8. Jun 14, 2005 #7

    OlderDan

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    Good point. After a long period of doing other things my introduction to these calculators the students all now have has been fairly recent. Punching in a 3 by 3 and hitting the T and -1 buttons is now such a trivial exercise I didn't even think of doing it by hand :smile:
     
  9. Jun 14, 2005 #8

    arildno

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    Calculators??
    Are those the things with frills and pink ribbons about them?
    I don't like that..
     
  10. Jun 14, 2005 #9
    Hi

    I got the correct result now.

    Thanks for Your answers,

    /Fred

     
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