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10.3: La Matriz Exponencial como Suma de Poderes
https://espanol.libretexts.org/Matematicas/Algebra_lineal/Libro%3A_An%C3%A1lisis_Matriz_(Cox)/10%3A_La_Matriz_Exponencial/10.03%3A_La_Matriz_Exponencial_como_Suma_de_Poderes
$(At)^k = \begin{pmatrix} {t^k}&{0}\\ {0}&{(2t)^k} \end{pmatrix} \nonumber$ \[e^{At} = \begin{pmatrix} {1-\frac{t^2}{2}+\frac{t^4}{4}+\cdots}&{t-\frac{t^3}{3!}+\frac{t^5}{5!}-\cdots}\\ {-t+\frac{t^3... $(At)^k = \begin{pmatrix} {t^k}&{0}\\ {0}&{(2t)^k} \end{pmatrix} \nonumber$ $e^{At} = \begin{pmatrix} {1-\frac{t^2}{2}+\frac{t^4}{4}+\cdots}&{t-\frac{t^3}{3!}+\frac{t^5}{5!}-\cdots}\\ {-t+\frac{t^3}{3!}-\frac{t^5}{5!}+\cdots}&{1-\frac{t^2}{2}+\frac{t^4}{4}+\cdots} \end{pmatrix} = \begin{pmatrix} {\cos(t)}&{\sin(t)}\\ {-\sin(t)}&{\cos(t)} \end{pmatrix} \nonumber$ $A^2 = A^3 = A^k = \begin{pmatrix} {0}&{1}\\ {0}&{0} \end{pmatrix} \nonumber$

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