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Writing a matrix as sum of a constant * matrix

  1. Apr 23, 2012 #1
    Ok I need some help on how I can approach a problem like this.

    Say that A B and C are matircies and I know the values of each of them.

    And let x and y be constants (ie: 7 or 2 or somthing like that)

    What I want to know is can you write

    A = xB + yC

    How would you check to see if there are an x and y that can make that equation true and if there is how can you find it. I think if B and C are in the span of A then yes there is an x and y but that doest tell me how to find them.
     
  2. jcsd
  3. Apr 23, 2012 #2
    You can check if there are [itex]x[/itex] and [itex]y[/itex] that make the equation true by solving the system of equations generated by matrices.

    Let matrix A be [itex]A=\left| \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right|[/itex], and B [itex]B=\left| \begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array} \right|[/itex]. Matrix C will be [itex]C=\left| \begin{array}{cc} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array} \right|[/itex].

    You get the system of equations to solve for [itex]x[/itex] and [itex]y[/itex] with the following expression:

    [itex]\left| \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right|[/itex] [itex]-[/itex] [itex]x \left| \begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array} \right|[/itex] [itex]-[/itex] [itex]y \left| \begin{array}{cc} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array} \right|[/itex] [itex]=[/itex] [itex]\left| \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right|[/itex].
     
  4. Apr 23, 2012 #3
    I was thinking of that as well the only problem is what if A B and C are 10 x 10 it would work but it is alot of work. Is there a more effcient method.
     
  5. Apr 23, 2012 #4
    Well, you need just a few equations to solve the system, so it doesn't matter if a matrix is 10 x 10. It wouldn't take too long. As far as I know, there is no more efficient method. Today, if we want to deal with 100 x 100 matrices, we use computers. That's why we've made them in the first place. :-)
     
  6. Apr 23, 2012 #5
    in your first post how would you expand that to a system of 2 eqns and 2 unkwns. The 2x2 constant matrix is what is screwing me up
     
  7. Apr 23, 2012 #6

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    You don't expand an equation with two by two matrices into a system of two equations. An equation involving two by two matrices is equivalent to a system of four equations, one for each of the four entries in the matrices. In general, you cannot solve an equation like "A= xB+ yC" for number x and y given matrices A, B, and C unless A, B, and C are carefully chosen, just as you cannot solve four equations in two unknowns unless the equations are "dependent". In general, there simply do NOT exist such numbers.
     
  8. Apr 24, 2012 #7
    Isn't the solution of the system we get an indicator of the existence of those numbers? If the system has a solution, such [itex]x[/itex] and [itex]y[/itex] exist. Otherwise, they don't.
     
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