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Can I use inverseof matrix?

  1. Sep 10, 2009 #1
    Hello all,

    Say I had n equations, with n variables, so that
    C1 = C1(f1), C1 is a funtion of f1 only
    C2 = C2(f1, f2), C2 is a function of f1 and f2
    ...
    Cn = Cn(f1, f2...fn), Cn is a function of all n vairables

    I can calculate the matrix dC/ df where each line is the derivative of Ci with respect to the n vairables, The matrix will look something lile

    a 0 0 0... 0
    b c 0 0... 0
    ...
    u v w x... z

    Now if I was after the matrix df/dC then am I basically just trying to invert this matrix?

    Thanks very much for any help
     
  2. jcsd
  3. Sep 10, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Yes, and, for a triangular matrix, that is fairly simple.
     
  4. Sep 10, 2009 #3
    Great. Thanks very much. Is there a particular algorithm that is recommended for a triangular matrix?

    Regards,
     
  5. Sep 11, 2009 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Just "back substitution". If your matrix is, say,
    [tex]\begin{bmatrix}a & 0 & 0 \\ b & c & 0\\ d & e & f\end{bmatrix}[/tex]
    then its inverse will be a matrix of the form
    [tex]\begin{bmatrix}u & 0 & 0 \\ v & w & 0\\ x & y & z\end{bmatrix}[/tex]
    such that
    [tex]\begin{bmatrix}a & 0 & 0 \\ b & c & 0\\ d & e & f\end{bmatrix}\begin{bmatrix}u & 0 & 0 \\ v & w & 0\\ x & y & z\end{bmatrix}= \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}[/tex]
    You must have au= 1 so u= 1/a. Then bu+ cv= b/a+ cv= 0 so v= -b/ac. cw= 1 so w= 1/c, etc.
     
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