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Math Help - Finding a basis

  1. #1
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    Finding a basis

    Let V be the real vector space of all 33 hermitian matrices with complex entries. Find a basis for V.
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  2. #2
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    Quote Originally Posted by iheartmathrookie View Post
    Let V be the real vector space of all 33 hermitian matrices with complex entries. Find a basis for V.
    The real vector space? Then a basis consists of \begin{bmatrix}1 & 0 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}, \begin{bmatrix}i & 0 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}, \begin{bmatrix}0 & i & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}, etc. for a total of 18 "basis" matrices.
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  3. #3
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    Let E_{ij} be the matrix with a 1 in the ij position and 0 elsewhere.

    Can you show

    E_{11} , E_{22} , E_{33}

    E_{12}+ E_{21}

    E_{13} + E_{31}

      E_{23} + E_{32}

      iE_{12}-iE_{21}

      iE_{13}-iE_{31}

      iE_{23}-iE_{32}


    form a basis?
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  4. #4
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    Quote Originally Posted by HallsofIvy View Post
    The real vector space? Then a basis consists of \begin{bmatrix}1 & 0 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}, \begin{bmatrix}i & 0 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}, \begin{bmatrix}0 & i & 0 \\0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}, etc. for a total of 18 "basis" matrices.

    Hermitian matrices have real diagonal entries and equal their transpose conjugates, so with this restriction we are left with a (9 dimensional) subspace of the real 3 x 3 matrices.
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  5. #5
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    Right. For some reason, my eyes passed right over the word "Hermitian"!
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