What is $e^{At}$?

Question

Currently, I only know how to solve systems of differential equations the eigenvector and eigenvalue way. So, one day, I decided to try something: $$\text{Because }\vec{x}' = A\vec{x}, $$ $$\text{I solved the "differential equation" and got } \vec{x} = e^{At}.$$ What does $e^{At}$ mean?

Answer 1

Generally, in order to "extend" functions to other than real numbers, we use the "Taylor series". In particular, $e^t= \sum \frac{1}{n!}t^n$. So we can define $e^{At}= \sum \frac{1}{n!}A^nt^n$