Question

How to be a cool kid and use a matrix as an exponent with an application. Probably best understood if you've taken Calculus II and Linear Algebra and are familiar with Rotation Matrices and the formula e^(ix). Let's rock and roll dudes.

Answer 1

This matrix is called the "generator" of rotations in the xy plane. \[\Large A = \left[\begin{matrix}0 & -1 \\ 1& 0\end{matrix}\right] \] we can multiply this by a scalar that tells us how much we want to rotate, we'll call it theta, cause that's what normal people seem to do.

So how do we get rotations out of this? We exponentiate it, like this: \[\huge e^{\theta A}\]

What does this even mean?

Answer 2

Well fortunately we can rewrite e^x as a power series! \[\Large e^{\theta A} = \sum_{n=0}^\infty \frac{\theta^n }{n!}A^n\] theta and n are scalars, but A is indeed a matrix.

Here are the first few from n=0 to n=2:

\[\Large A^0 = I \\ \Large A^1 = A \\ \Large A^2 = -I\]

Someone calculate the powers of A from n=3 up to n=5 and I'll keep going and reveal the "cool part". ;)