Results 1 to 5 of 5

Math Help - Basis for matrices of trace 0

  1. #1
    Newbie
    Joined
    Sep 2010
    Posts
    15

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member MacstersUndead's Avatar
    Joined
    Jan 2009
    Posts
    291
    Thanks
    32
    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
    \mathrm{tr}(A) = a_{11} + a_{22} + \dots + a_{nn}=\sum_{i=1}^{n} a_{i i}

    If \mathrm{tr}(A) = 0 then all 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

    <br />
\left[ {\begin{array}{cc}<br />
 0 & 1  \\<br />
 0 & 0  \\<br />
 \end{array} } \right] <br />
and <br />
\left[ {\begin{array}{cc}<br />
 0 & 0  \\<br />
 1 & 0  \\<br />
 \end{array} } \right]<br />

    EDIT:// ** with  i \neq j
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2010
    Posts
    15
    But if the 1,1th entry is 1, and the 2,2th entry is -1, we also have a zero trace.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member MacstersUndead's Avatar
    Joined
    Jan 2009
    Posts
    291
    Thanks
    32
    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.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,405
    Thanks
    1328
    A basis consists of all n by n matrices in which
    1) For i and j from 1 to n with i\ne j, A_{ij} with entries a_{ij}= 1 and all other entries 0 and
    2) For i and j from 1 to n with j> i, B_{ij} with entries b_{ii}= 1 and b_{jj}= -1.
    and all other entries 0.
    If n= 2, those would be B_{12}= \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}, A_{12}= \begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}, and A_{21}= \begin{bmatrix}0 & 0 \\ 1 & 0\end{bmatrix}.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] M 2x2 matrices of C with standard basis.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 30th 2011, 12:25 AM
  2. [SOLVED] Linear Algebra, trace similar matrices
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 22nd 2010, 11:43 AM
  3. Basis and dimension of Matrices
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 15th 2009, 09:30 AM
  4. Matrices- Trace
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 22nd 2009, 03:06 PM
  5. Basis Matrices
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 1st 2008, 12:22 PM

Search Tags


/mathhelpforum @mathhelpforum