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Can a 2N by 2N matrix written in terms of N by N matrices?

  1. Oct 4, 2013 #1
    I posted this question over at the QM page,

    https://www.physicsforums.com/showthread.php?t=714076

    but I realized I am really looking for a

    hard Mathematical proof ...

    A description of a numerical way of proving this would also be very helpful for me.

    or a reference covering the subject.

    Many thanks in advance,
     
  2. jcsd
  3. Oct 4, 2013 #2

    fzero

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    As a warmup, would you know how to prove that ##\{ I_{2}, \sigma_i\}## is a basis for Hermitian ##2\times 2## matrices? The result for ##2N\times 2N## will follow by writing the matrix in block form and using the basis as explained by wle in that thread.
     
  4. Oct 4, 2013 #3
    Yes - I can do the 2x2 proof I guess.

    Because any 2x2 Hermitian matrix can be written as:

    [tex]
    H=\begin{bmatrix}
    a & c -i \ d \\
    c + i \ d & b
    \end{bmatrix}
    [/tex]

    where a,b,c,d are all real numbers.

    Then H can be uniquely defined in terms of Pauli matrices:
    [tex]
    \frac{1}{2}\left[ (a+b) \ I_{2\times 2} + (a-b) \ \sigma_z + 2 \ c \ \sigma_x + 2 \ d \ \sigma_y\right]
    [/tex]

    But how to extend this to 2N by 2N ?
     
  5. Oct 4, 2013 #4
    Yes, I got it ...

    Just write it out explicitly and choose A,B,C,D accordingly to get the random 2N by 2N matrix.

    Many thanks for directing me to that.
     
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