$$$e^{\left[\begin{array}{cc}t & - t\\0 & t\end{array}\right]}$$$

Halla $$$e^{\left[\begin{array}{cc}t & - t\\0 & t\end{array}\right]}$$$.

Primero, diagonalice la matriz (para ver los pasos, consulte calculadora de diagonalización de matrices).

Como la matriz no es diagonalizable, exprésala como la suma de la matriz diagonal $$$D = \left[\begin{array}{cc}t & 0\\0 & t\end{array}\right]$$$ y la matriz nilpotente $$$N = \left[\begin{array}{cc}0 & - t\\0 & 0\end{array}\right]$$$.

Nótese que $$$N^{2} = \left[\begin{array}{cc}0 & 0\\0 & 0\end{array}\right]$$$.

Esto significa que $$$e^{N} = I + N$$$, es decir, $$$e^{\left[\begin{array}{cc}0 & - t\\0 & 0\end{array}\right]} = \left[\begin{array}{cc}1 & 0\\0 & 1\end{array}\right] + \left[\begin{array}{cc}0 & - t\\0 & 0\end{array}\right] = \left[\begin{array}{cc}1 & - t\\0 & 1\end{array}\right].$$$

La exponencial de una matriz diagonal es una matriz cuyas entradas diagonales están exponentiadas: $$$e^{\left[\begin{array}{cc}t & 0\\0 & t\end{array}\right]} = \left[\begin{array}{cc}e^{t} & 0\\0 & e^{t}\end{array}\right].$$$

Ahora, $$$e^{\left[\begin{array}{cc}t & - t\\0 & t\end{array}\right]} = e^{\left[\begin{array}{cc}t & 0\\0 & t\end{array}\right] + \left[\begin{array}{cc}0 & - t\\0 & 0\end{array}\right]} = e^{\left[\begin{array}{cc}t & 0\\0 & t\end{array}\right]}\cdot e^{\left[\begin{array}{cc}0 & - t\\0 & 0\end{array}\right]} = \left[\begin{array}{cc}e^{t} & 0\\0 & e^{t}\end{array}\right]\cdot \left[\begin{array}{cc}1 & - t\\0 & 1\end{array}\right].$$$

Finalmente, multiplica las matrices:

$$$\left[\begin{array}{cc}e^{t} & 0\\0 & e^{t}\end{array}\right]\cdot \left[\begin{array}{cc}1 & - t\\0 & 1\end{array}\right] = \left[\begin{array}{cc}e^{t} & - t e^{t}\\0 & e^{t}\end{array}\right]$$$ (para ver los pasos, vea calculadora de multiplicación de matrices).

$$$e^{\left[\begin{array}{cc}t & - t\\0 & t\end{array}\right]} = \left[\begin{array}{cc}e^{t} & - t e^{t}\\0 & e^{t}\end{array}\right]$$$A