Homework Help: A question in prooving liniar operator

1. Mar 7, 2008

2. Mar 7, 2008

HallsofIvy

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
$$\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]^T= \left[\begin{array}{cc} a & c \\ b & d\end{array}\right]$$
so that
$$\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]^T- \left[\begin{array}{cc}a & b \\ c & d\end{array}\right]= \left[\begin{array}{c c} a & c \\ b & d\end{array}\right]- \left[\begin{array}{cc}a & b \\ c & d\end{array}\right][/itex] [tex]= \left[\begin{array}{cc} 0 & c- b \\ b- c & 0\end{array}\right]$$

Last edited by a moderator: Mar 7, 2008
3. Mar 7, 2008

transgalactic

no this sign is a little line like a derivative sign
no way it cant be a T
??

4. Mar 8, 2008

HallsofIvy

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.

5. Mar 8, 2008

transgalactic

i trust your judgement probably its a print mistake
and they ment T

6. Mar 8, 2008

HallsofIvy

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 by a moderator: Mar 8, 2008