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It is an easily proved fact that for a $2\times 2$ traceless matrix $A$,

$$ e^A = \cos\left(\sqrt{\det(A)}\right)I + \frac{\sin\left(\sqrt{\det(A)}\right)}{\sqrt{\det(A)}}A$$

Problem 2.7 of Lie Groups, Lie Algebras, and Representations by Bryan Hall asks to use this fact to compute $\exp(X)$, where

$$ X = \begin{pmatrix} 4 & 3\\ -1 & 2 \end{pmatrix}$$

In other words, I have to write $X$ in terms of traceless matrices, and employ the above fact. My question is: is there a systematic way to do this?

My idea to solve this problem is to write $X = X_1 + X_2$, where $X_1$ is traceless, $X_2$ is diagonal or nilpotent, and $[X_1, X_2] = 0$, and compute the exponent using $e^{X_1 + X_2} = e^{X_1}e^{X_2}$. For example, I tried the most obvious thing: $$X = \begin{pmatrix} -2 & 3\\-1 & 2\end{pmatrix} + \begin{pmatrix} 6 & 0\\0 & 0\end{pmatrix},$$

but the two matrices above do not commute.

Rodrigo de Azevedo
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Solarflare0
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1 Answers1

2

You were close.

Try $X=X_1+X_2$, where $X_1=X-3I$ and $X_2=3I$.

quasi
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  • Ah I see, you take $((X_{11} + X_{22})/2)I$ as the matrix you remove from $X$, and this commutes since it is a multiple of the identity. Thanks! – Solarflare0 Jun 14 '21 at 00:05