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

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

Pertama, lakukan diagonalisasi pada matriks tersebut (untuk langkah-langkahnya, lihat matrix diagonalization calculator).

Karena matriks tersebut tidak dapat didiagonalkan, tuliskan sebagai jumlah dari matriks diagonal $$$D = \left[\begin{array}{cc}t & 0\\0 & t\end{array}\right]$$$ dan matriks nilpoten $$$N = \left[\begin{array}{cc}0 & - t\\0 & 0\end{array}\right]$$$.

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

Ini berarti bahwa $$$e^{N} = I + N$$$, yaitu $$$e^{\left[\begin{array}{cc}0 & - t\\0 & 0\end{array}\right]} = \left[\begin{array}{cc}1 & 0\\0 & 1\end{array}\right] + \left[\begin{array}{cc}0 & - t\\0 & 0\end{array}\right] = \left[\begin{array}{cc}1 & - t\\0 & 1\end{array}\right].$$$

Eksponensial dari suatu matriks diagonal adalah matriks yang entri-entri diagonalnya dieksponensialkan: $$$e^{\left[\begin{array}{cc}t & 0\\0 & t\end{array}\right]} = \left[\begin{array}{cc}e^{t} & 0\\0 & e^{t}\end{array}\right].$$$

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

Terakhir, kalikan matriks-matriks tersebut:

$$$\left[\begin{array}{cc}e^{t} & 0\\0 & e^{t}\end{array}\right]\cdot \left[\begin{array}{cc}1 & - t\\0 & 1\end{array}\right] = \left[\begin{array}{cc}e^{t} & - t e^{t}\\0 & e^{t}\end{array}\right]$$$ (untuk langkah-langkahnya, lihat kalkulator perkalian matriks).

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