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

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

İlk olarak, matrisi köşegenleştirin (adımlar için bkz. matrix diagonalization calculator).

Matris diyagonalleştirilemez olduğundan, onu diyagonal matris $$$D = \left[\begin{array}{cc}t & 0\\0 & t\end{array}\right]$$$ ile nilpotent matris $$$N = \left[\begin{array}{cc}0 & - t\\0 & 0\end{array}\right]$$$'nin toplamı olarak yazın.

Dikkat edin ki $$$N^{2} = \left[\begin{array}{cc}0 & 0\\0 & 0\end{array}\right]$$$.

Bu, $$$e^{N} = I + N$$$ anlamına gelir, yani $$$e^{\left[\begin{array}{cc}0 & - t\\0 & 0\end{array}\right]} = \left[\begin{array}{cc}1 & 0\\0 & 1\end{array}\right] + \left[\begin{array}{cc}0 & - t\\0 & 0\end{array}\right] = \left[\begin{array}{cc}1 & - t\\0 & 1\end{array}\right].$$$

Diyagonal bir matrisin üsseli, köşegen öğelerinin üssü alınmış bir matristir: $$$e^{\left[\begin{array}{cc}t & 0\\0 & t\end{array}\right]} = \left[\begin{array}{cc}e^{t} & 0\\0 & e^{t}\end{array}\right].$$$

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

Son olarak, matrisleri çarpın:

$$$\left[\begin{array}{cc}e^{t} & 0\\0 & e^{t}\end{array}\right]\cdot \left[\begin{array}{cc}1 & - t\\0 & 1\end{array}\right] = \left[\begin{array}{cc}e^{t} & - t e^{t}\\0 & e^{t}\end{array}\right]$$$ (adımlar için bkz. matris çarpımı hesaplayıcısı.)

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