exp of a matrix


by Jhenrique
Tags: matrix
Jhenrique
Jhenrique is offline
#1
Feb12-14, 03:24 AM
P: 419
Is correct my step by step below?

[tex]
\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}[/tex]
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tiny-tim
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#2
Feb12-14, 04:13 AM
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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?
D H
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#3
Feb12-14, 05:40 AM
Mentor
P: 14,435
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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