Powers of a Toeplitz matrix

Question

I'm searching a closed formula to compute the powers of the following matrix \begin{equation*}F\triangleq \begin{bmatrix} 1 & T & \frac{T^2}{2}\\ 0 & 1 & T\\ 0 & 0 & 1 \end{bmatrix}\end{equation*} where $T$ is a given parameter. For example, if I'm not wrong the first 5+1 powers are \begin{equation*}F^0\triangleq \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}\end{equation*} \begin{equation*}F^1\triangleq \begin{bmatrix} 1 & T & \frac{T^2}{2}\\ 0 & 1 & T\\ 0 & 0 & 1 \end{bmatrix}\end{equation*} \begin{equation*}F^2\triangleq \begin{bmatrix} 1 & 2T & 2T^2\\ 0 & 1 & 2T\\ 0 & 0 & 1 \end{bmatrix}\end{equation*} \begin{equation*}F^3\triangleq \begin{bmatrix} 1 & 3T & \frac{9T^2}{2}\\ 0 & 1 & 3T\\ 0 & 0 & 1 \end{bmatrix}\end{equation*} \begin{equation*}F^4\triangleq \begin{bmatrix} 1 & 4T & 8T^2\\ 0 & 1 & 4T\\ 0 & 0 & 1 \end{bmatrix}\end{equation*} \begin{equation*}F^5\triangleq \begin{bmatrix} 1 & 5T & \frac{25T^2}{2}\\ 0 & 1 & 5T\\ 0 & 0 & 1 \end{bmatrix}\end{equation*} From these results it seems to me that for any $k$ holds \begin{equation*}F^k=\begin{bmatrix} 1 & kT & \frac{(kT)^2}{2} \\ 0 & 1 & kT \\ 0 & 0 & 1 \end{bmatrix}\end{equation*} is it true? If yes, how can be proved my formula?

Answer 1

Your matrix can be written as an exponential of a nilpotent matrix: $$F=e^{TN},\qquad N=\left(\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{array}\right).$$ Note that $N^2=\left(\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)$ and $N^3=0$. Now, $$F^k=e^{kT N}=\mathbf 1+kTN+\frac{k^2T^2}{2}N^2,$$ which gives the result.

P.S. If you are uncomfortable with matrix exponentials, then simply use induction.