Matrix powers limitations.

[lalala_land89]

Okay i'm supposed to explore matrix powers pattern find a scope or limitations to it.
N is the power
k is any real number
|k+1 K-1|^N
|k-1 k+1|

I found a pattern |(2^n-1)*(k^2+1) (2^n-1)*(k^2-1)|
........................ ... |(2^n-1)*(k^2-1) (2^n-1)*(k^2+1)|

so i need to look at further values of k and N and state the scope or limitations of k and N. i did that and i dont see any limitations. help plz?

 

[soroban]

Hello, lalala_land89!

Your pattern is a bit off . . .

I'm supposed to explore matrix powers patterns.
Find a scope or limitations to it.

\(\displaystyle N\) is the power, \(\displaystyle k\) is any real number.

\(\displaystyle \L A \;=\;\begin{pmatrix}k+1\;\; & k-1 \\ \quad & \quad \\ k-1\;\; & k+1\end{pmatrix}^N\)

The pattern is: \(\displaystyle \L\:A^{^N}\;=\;2^{^{N-1}}\begin{pmatrix}k^{^N} + 1\;\; & k^{^N} - 1 \\ \quad & \quad \\ k^{^N} - 1\;\; & k^{^N} + 1\end{pmatrix}\)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

An observation . . .


If \(\displaystyle \,|k|\,<\,1\), then: \(\displaystyle \:\lim_{N\to\infty}\left(k^N\right) \,=\,0\)


Hence, for large \(\displaystyle N:\L\;\;A^{^N}\; \rightarrow\;\;2^{^{N-1}}\begin{pmatrix}1\;& -1 \\ -1\;& 1\end{pmatrix}\)