I have to calulate A^10.
I have been given A =
(1 2
2 1)
and B =
(1 1
1 -1)
I have calculated that B^-1AB =
(3 0
0 1)
and I don't really understand what my notes mean to take it on from here. Help would be appreciated.
To find $\displaystyle A^{10}$
You must find a diagonal matrix $\displaystyle D$ such that
$\displaystyle A^n= BD^nB^{-1} \implies D^n= B^{-1}A^nB$
In your case $\displaystyle n=10$
and where $\displaystyle B$ is the eigenvectors of $\displaystyle A$
So first thing to do is find the eigenvalues and eigenvectors of $\displaystyle A$
It looks like you have been given that so use
$\displaystyle A^n= BD^nB^{-1} $
As you have $\displaystyle D , B$ and $\displaystyle B^{-1}$
If that was a little to hard to follow then,
$\displaystyle
\left( {\begin{array}{*{20}{c}}
1& 2 \\
2 & 1\\
\end{array}} \right) ^{10}
=
\left( {\begin{array}{*{20}{c}}
1 & 1 \\
{-1} & 1 \\
\end{array}} \right)
\left( {\begin{array}{*{20}{c}}
3 & 0 \\
0 & 1 \\
\end{array}} \right) ^{10}
\left( {\begin{array}{*{20}{c}}
1 & 1 \\
{-1} & 1 \\
\end{array}} \right) ^{-1}
$
Just solve the RHS.