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A question in prooving liniar operator

  1. Mar 7, 2008 #1
    http://img228.imageshack.us/my.php?image=52788065ff7.jpg

    the formula which i was given is fro a polinomial
    although the operator is defined for normal 2X2 vector

    i dont know how to aproach to this formula???
     
  2. jcsd
  3. Mar 7, 2008 #2

    HallsofIvy

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    I'm not sure what you mean by "for a polynomial". The formula is T(A)= AT- A.

    Perhaps you are thinking that AT is a power? Even if it were, that would still be alright- if you can multiply matrices you can certainly take a matrix to a power- and a product of matrices still represents a linear transformation.

    However, AT is the standard notation for the 'transpose' of a matrix: basically you swap rows and columns. For a 2 by 2 matrix
    [tex]\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]^T= \left[\begin{array}{cc} a & c \\ b & d\end{array}\right][/tex]
    so that
    [tex]\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]^T- \left[\begin{array}{cc}a & b \\ c & d\end{array}\right]= \left[\begin{array}{c c} a & c \\ b & d\end{array}\right]- \left[\begin{array}{cc}a & b \\ c & d\end{array}\right][/itex]
    [tex]= \left[\begin{array}{cc} 0 & c- b \\ b- c & 0\end{array}\right][/tex]
     
    Last edited: Mar 7, 2008
  4. Mar 7, 2008 #3
    no this sign is a little line like a derivative sign
    no way it cant be a T
    ??
     
  5. Mar 8, 2008 #4

    HallsofIvy

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    then I have no idea what it means- it's not a standard notation. Does your book give a definition? If not, ask your teacher.
     
  6. Mar 8, 2008 #5
    i trust your judgement probably its a print mistake
    and they ment T
     
  7. Mar 8, 2008 #6

    HallsofIvy

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    It's my vision I'm not sure you should trust! Is this matrix over the real numbers or complex numbers? Sometimes a little "sword" superscript is use to represent the "Hermitian conjugate" where you take the transpose (switch rows to columns) and take the complex conjugate of all entries. Of course, if your matrix has only real number entries, that is the same as the transpose.
     
    Last edited: Mar 8, 2008
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