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pondzo
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Homework Statement
Compute ##A^j~\text{for} ~~j=1,2,...,n## for the block diagonal matrix##A=\begin{bmatrix}
J_2(1)& \\
&J_3(0)
\end{bmatrix}##,
And show that the difference equation ##x_{j+1}=Ax_{j}## has a solution satisfying ##|x_{j}|\rightarrow\infty~\text{as}~j\rightarrow\infty##
Homework Equations
The Attempt at a Solution
So ##A^1=\begin{bmatrix}1&1&0&0&0 \\
0&1&0&0&0 \\
0&0&0&1&0 \\
0&0&0&0&1 \\
0&0&0&0&0
\end{bmatrix}
,~~A^2=\begin{bmatrix}
1&2&0&0&0 \\
0&1&0&0&0 \\
0&0&0&0&1 \\
0&0&0&0&0 \\
0&0&0&0&0
\end{bmatrix}
,~~A^j=\begin{bmatrix}
1&j&0&0&0 \\
0&1&0&0&0 \\
0&0&0&0&0 \\
0&0&0&0&0 \\
0&0&0&0&0
\end{bmatrix}\forall ~~j\geq 3 ##
I am certain that this isn't a difficult question, but I am not sure how to apply this to the difference equation. Which is probably due to my lack of experience with them. Help would be appreciated, thanks.
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