Euler identity

Another interesting paper came out last week on the preprint archive, “A matrix generalization of Euler identity.” The Euler identity reads,

$e^{jx} = \cos(x) + j \sin(x)$

and is traditionally derived from the Taylor expansion of sine and cosine. In this paper, the author generalizes the Euler identity to matrices. He does this by introducing a complex matrix, known as the imaginary unit matrix

$\mathbf{\Phi} = \left( \begin{array}{cc} 0 & j \\j \alpha^2 & 0\end{array} \right)$

Just like j^2 = -1 , $\mathbf{\Phi^2} = - \alpha^2 \mathbf{I}$ .

He then proves the Euler identity for these matrices as:

$e^{x \mathbf{\Phi}} = \cos(\alpha x)\mathbf{I} + \frac{1}{\alpha} \sin(\alpha x) \mathbf{\Phi}$

for all real x.

Possibly related:

Numerical Python
Variational Integrators
Higher order functions
Fluid Match
Random Facts

This entry was posted on Saturday, March 17th, 2007 at 5:52 pm and is filed under Paper. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

Ganesh Swami

Euler identity

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