# exp of a matrix

by Jhenrique
Tags: matrix
 P: 279 Is correct my step by step below? \begin{aligned} \exp \left (\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{bmatrix} \right ) &= \exp \left ( a_{11} \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix} + a_{22} \begin{bmatrix} 0 & 0 \\ 0 & 1 \\ \end{bmatrix} + a_{12} \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix} + a_{21} \begin{bmatrix} 0 & 0 \\ 1 & 0 \\ \end{bmatrix} \right ) \\ & = \exp\left ( \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix} \right )^{a_{11}} \exp\left ( \begin{bmatrix} 0 & 0 \\ 0 & 1 \\ \end{bmatrix} \right )^{a_{22}} \exp\left ( \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix} \right )^{a_{12}} \exp\left ( \begin{bmatrix} 0 & 0 \\ 1 & 0 \\ \end{bmatrix} \right )^{a_{21}} \\ &= \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^{a_{11}} \begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix}^{a_{22}} \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}^{a_{12}} \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}^{a_{21}} \end{aligned}
 Sci Advisor HW Helper Thanks PF Gold P: 26,126 Hi Jhenrique! Hint: i] what is exp##\begin{bmatrix} a & 0 \\ 0 & a \\ \end{bmatrix}## ? what is exp##\begin{bmatrix} 0 & a \\ a & 0 \\ \end{bmatrix}## ? ii] do your final matrices commute?
 Mentor P: 14,243 Jhenrique, I fixed your LaTeX so it doesn't spill across the screen. The answer to your question is no. You made multiple errors. Your second line erroneously assumes ##\exp(A+B) = \exp(A)\exp(B)## and also erroneously assumes ##\exp(sA)=\exp(A)^s##.

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