# hermitian matrices, pauli matrices

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!

• What is the $M$ in $\exp(MA_i)$ (or exp(MAi))? – Lord Soth Apr 17 '13 at 17:24
• M is 2x2 matrix. Thanks – xxxx Apr 17 '13 at 17:30
• I mean e^(M*Ai) where i=1,2,3 – xxxx Apr 17 '13 at 17:30
• So $M$ can be an arbitrary $2\times 2$ matrix, or is it positive definite, etc? – Lord Soth Apr 17 '13 at 17:33
• Let's assume that is arbitrary matrix – xxxx Apr 17 '13 at 17:41

## 1 Answer

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.