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Thread: Basis for matrices of trace 0

  1. #1
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    Basis for matrices of trace 0

    Hey, I was wondering if someone could help me find a basis for the space of all nxn matricies, over R, with trace 0.
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  2. #2
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    Quote Originally Posted by HelloWorld2 View Post
    Hey, I was wondering if someone could help me find a basis for the space of all nxn matricies, over R, with trace 0.
    It's been a long while, but maybe I can give a small push. Take this with a grain of salt

    We know that
    $\displaystyle \mathrm{tr}(A) = a_{11} + a_{22} + \dots + a_{nn}=\sum_{i=1}^{n} a_{i i}$

    If $\displaystyle \mathrm{tr}(A) = 0$ then all $\displaystyle a_{i i} = 0$.

    A basis for the space of all nxn matricies, then, is the set of all matricies with 1 as an ij-th entry **, and all the rest being 0. go through all the possibilities.

    ex. for a 2x2 matrix, the basis would be

    $\displaystyle
    \left[ {\begin{array}{cc}
    0 & 1 \\
    0 & 0 \\
    \end{array} } \right]
    $ and $\displaystyle
    \left[ {\begin{array}{cc}
    0 & 0 \\
    1 & 0 \\
    \end{array} } \right]
    $

    EDIT:// ** with $\displaystyle i \neq j$
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  3. #3
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    But if the 1,1th entry is 1, and the 2,2th entry is -1, we also have a zero trace.
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  4. #4
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    Quote Originally Posted by HelloWorld2 View Post
    But if the 1,1th entry is 1, and the 2,2th entry is -1, we also have a zero trace.
    oh yeah, sorry. what a silly mistake!

    EDIT:// now I'm wondering how you would generalize, b/c I think the form where you have a and -a on the diagonal, replacing the entries for the previous example basis' would be sufficient for 2x2 matricies.
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  5. #5
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    A basis consists of all n by n matrices in which
    1) For i and j from 1 to n with $\displaystyle i\ne j$, $\displaystyle A_{ij}$ with entries $\displaystyle a_{ij}= 1$ and all other entries 0 and
    2) For i and j from 1 to n with $\displaystyle j> i$, $\displaystyle B_{ij}$ with entries $\displaystyle b_{ii}= 1$ and $\displaystyle b_{jj}= -1$.
    and all other entries 0.
    If n= 2, those would be $\displaystyle B_{12}= \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$, $\displaystyle A_{12}= \begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}$, and $\displaystyle A_{21}= \begin{bmatrix}0 & 0 \\ 1 & 0\end{bmatrix}$.
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