# Diagonalizable matrix used in polynomial form

• Mar 30th 2011, 10:48 AM
LHS
Diagonalizable matrix used in polynomial form
I'm rather unsure what is going on with the last part of question c any help would be greatly appreciated!

http://img163.imageshack.us/img163/4992/photo1mpl.jpg
• Mar 30th 2011, 10:59 AM
FernandoRevilla
Quote:

Originally Posted by LHS
I'm rather unsure what is going on with the last part of question c any help would be greatly appreciated!

http://img846.imageshack.us/img846/2817/photoxtk.jpg

If you don`t rotate the problem, we'll suffer from neck ache.
• Mar 30th 2011, 11:12 AM
LHS
sorry.. was suffering from some technical difficulties!
• Mar 31st 2011, 02:12 PM
masnarski
Hi there,

Your matrix is diagonalizable. In particular,

$
B=PDP^{-1}, where
D=\begin{pmatrix}
6 & 0 & 0 \\
0 & 6 & 0 \\
0 & 0 & 3\\
\end{pmatrix}\\
$

Hence,

$
B^n = (PDP^{-1})^n = (PDP^{-1})(PDP^{-1}) ... (PDP^{-1})\\
= PD^nP^{-1}\\
$

Substituting this into your equation yields,

$

a_nPD^nP^{-1} + ... + a_1PDP^{-1} + a_0& = C\\

$

and pre and post multiplying both sides by P inv. and P (respectively) yields:

$

a_nD^n + ... + a_0& = P^{-1}CP\\

$

where C is that diagonal matrix you are given on the right-hand side of the equation (with entries 1, 2, 3).

However, you will notice that when you multiply P^{-1}CP out, you will get a matrix that is not diagonal on the right hand side of the last equation.

But a power of a diagonal matrix is again a diagonal matrix, a scalar multiple of a diagonal matrix is also diagonal, and the sum of diagonal matrices is again diagonal. Hence, you will never get the RHS, so it is impossible.
• Mar 31st 2011, 02:25 PM
LHS
Can't really thank you enough.. looks worryingly simple now! You know, one of those questions you have a mental block on, that was bothering the life out of me
• Mar 31st 2011, 07:02 PM
masnarski
Don't worry about it. It's something I might not have figured out some other day.

When I am facing problems like this, and a potential way of solving it doesn't occur to me within a few minutes, I consciously remind myself of "what I have in my toolbox." This involves not only summoning up what I know about diagonalizable matrices (and perhaps even making lists and drawing arrows to indicate potential connections between ideas), but also reminding myself of the basic methods of proof: direct proof, contrapositive, contradiction, etc. When in doubt -- proof by induction!
• Apr 1st 2011, 01:54 AM
LHS
That's good advice, i'll bear it in mind :) cheers mate!