Exponential of a Matrix

  • #1
Will anyone help me to find out the analytic expression
of the following [tex]2^N\times2^N[/tex] exponential?

[tex]exp[t(X\otimes X\otimes I\ldots\otimes I+I\otimes X\otimes X\otimes I\ldots\otimes I+\ldots+I\otimes I\otimes\ldots I \otimes X \otimes X+X\otimes I\ldots I\otimes X)][/tex],

where

[tex]
I= \left[\begin{array}{cc}
1 & 0 \\
0 & 1 \end{array}\right] [/tex]

and
[tex]
X=\left[\begin{array}{cc}
0 & 1 \\
1 & 0 \end{array}\right]

[/tex].

[Note that the parenthesis in the `exponential' contains sum of N+1 terms each of which is a tensor product of 2 Xs and (N-2) of Is in some order.]

I've evaluated (via Mathematica) for N=3,4,5,6. But I need an analytic expression for it.

Thanks and Regards.
 

Answers and Replies

  • #2
13,582
10,710
I'm not quite sure how you arrange the tensor product to get a matrix form again. You can work with a standard basis ##E_{ij} = \begin{cases} 1&\text{ at position }(i,j) \\0&\text{ elsewhere }\end{cases}##, and write the combined matrix accordingly. But this looks as if you want to calculate the derivative at ##t=0## for some purpose, in which case it would probably make more sense to work with $I,X$ where I would write ##X=P## as the permutation it is to make it more obvious.

Investigation of small ##n## and look for an induction should help, too.
 

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