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 November 1st, 2017, 04:12 PM #1 Member   Joined: Nov 2016 From: Kansas Posts: 73 Thanks: 1 Power Formula Power Formula for A=$\begin{pmatrix} \Lambda & 1 & 0\\ 0 & \Lambda & 1\\ 0 & 0 & 1 \end{pmatrix}^n$ Last edited by ZMD; November 1st, 2017 at 04:16 PM.
November 1st, 2017, 04:41 PM   #2
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Quote:
 Originally Posted by ZMD Power Formula for A=$\begin{pmatrix} \Lambda & 1 & 0\\ 0 & \Lambda & 1\\ 0 & 0 & 1 \end{pmatrix}^n$
I don't know what you mean by the term "power formula." But it's fairly easy to spot the pattern for $\displaystyle A^n$ by working out a few terms if that's what you need. I get
$\displaystyle A^n = \left ( \begin{matrix} \Lambda ^n & n \Lambda ^{n - 1} & 1 + 2 \Lambda + \dots + (n - 1) \Lambda ^{n - 2} \\ 0 & \Lambda ^n & 1 + \Lambda + \Lambda ^2 + \dots + \Lambda ^{n - 1} \\ 0 & 0 & 1 \end{matrix} \right )$

You can use induction to prove this formula.

-Dan

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