# Thread: Diagonalizable matrix used in polynomial form

1. ## 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!

2. 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!

If you don`t rotate the problem, we'll suffer from neck ache.

3. sorry.. was suffering from some technical difficulties!

4. 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.

5. 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

6. 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!

7. That's good advice, i'll bear it in mind cheers mate!