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These matrices are the Pauli matrices \begin{align} A_1 & = \left[\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right] \\ A_2 & = \left[\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right] \\ A_3 & = \left[\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right] \\ \end{align} Consider an arbitrary $2\times 2$ matrix $M$. Then, what are the matrix exponentials $\exp(MA_i)$ for $i=1,2,3$?

Thanks!

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  • $\begingroup$ What is the $M$ in $\exp(MA_i)$ (or exp(MAi))? $\endgroup$ – Lord Soth Apr 17 '13 at 17:24
  • $\begingroup$ M is 2x2 matrix. Thanks $\endgroup$ – xxxx Apr 17 '13 at 17:30
  • $\begingroup$ I mean e^(M*Ai) where i=1,2,3 $\endgroup$ – xxxx Apr 17 '13 at 17:30
  • $\begingroup$ So $M$ can be an arbitrary $2\times 2$ matrix, or is it positive definite, etc? $\endgroup$ – Lord Soth Apr 17 '13 at 17:33
  • $\begingroup$ Let's assume that is arbitrary matrix $\endgroup$ – xxxx Apr 17 '13 at 17:41
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For a 2x2 matrix $$ M=\begin{bmatrix} a&b \\ c& d \end{bmatrix} $$ the exponential is given by $$ e^M=e^{(a+d)/2}\begin{bmatrix} C_{abcd}+(a-d)S_{abcd} & 2bS_{abcd} \\ 2cS_{abcd} & C_{abcd}+(d-a)S_{abcd} \end{bmatrix} $$ where $$ C_{abcd} = \cosh\left(\frac{1}{2}\sqrt{4bc+(a-d)^2}\right) $$ and $$ S_{abcd} = \frac{\sinh\left(\frac{1}{2}\sqrt{4bc+(a-d)^2}\right)}{\sqrt{4bc+(a-d)^2}} $$ use these formulae with $$ MA_1=\begin{bmatrix} b&a \\ d& c \end{bmatrix} $$ $$ MA_2=i\begin{bmatrix} b&-a \\ d& -c \end{bmatrix} $$ $$ MA_3=\begin{bmatrix} a&-b \\ c&-d \end{bmatrix} $$ noting that $\sinh$ and $\cosh$ change to $\sin$ and $\cos$ with imaginary arguments.

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