What does $e^{-iAt}$ mean?

Question

I'm trying to understand an algorithm, to solve $Ax = b$ linear equations.

But there is an equation, which I can not understand: $e^{-iAt}$ What does it mean, to calculate the $-iAt$ power of $e$ ?

Answer 1

Use the matrix exponential:

$$ e^A = I + A + \frac12 A^2 + \frac 1{3!} A^3 + \cdots $$

You can check for yourself that it satisfies the important property:

$$ \frac{d}{dt} e^{tA} = Ae^{tA} = e^{tA}A $$

Answer 2

It is a little unclear, but for the $-iAt$ power of $e$, I'm going to assume that $i$ is the $\sqrt{-1}$ so that the expression is an combination of a real and an imaginary number as you can determine from the series expansion. Think of the real numbers as being on the the x-axis and the imaginary numbers being on the y-axis. Relationships using real and imaginary numbers happen in all sorts of scientific problems.