# Show this matrix is similar to a diagonal matrix

• Aug 2nd 2011, 03:46 AM
EuropeanSon
Show this matrix is similar to a diagonal matrix
A square matrix $\displaystyle A$ (of some size $\displaystyle n \times n$) satisfies the condition $\displaystyle A^2 - 8A +15I = 0.$

(a) Show that this matrix is similar to a diagonal matrix.

(b) Show that for every positive integer $\displaystyle k \geq 8$ there exists a matrix $\displaystyle A$ satisfying the above condition with $\displaystyle tr(A) = k$.

I have little or no idea where to go with this problem. I know that for (a), $\displaystyle A^2 - 8A$ is obviously a diagonal matrix $\displaystyle (-15I)$, but am not sure if this is of any use. I also am not sure if I can simply say $\displaystyle A^2 - 8A + 15I = 0$, so $\displaystyle (A - 5I) (A - 3I) = 0$, in which case, either (i) $\displaystyle A = 3I$ or $\displaystyle A = 5I$, or (ii) $\displaystyle (A - 5I) (A - 3I) = 0$ with $\displaystyle A - 5I \neq 0$ and $\displaystyle A -3I \neq 0$, and in this case, I'm not sure how to use (ii) to prove anything. Obviously (b) has something to do with the second term being -8A, but apart from this I can't really think of anything.

Any help would be greatly appreciated.
• Aug 2nd 2011, 04:38 AM
FernandoRevilla
Re: Show this matrix is similar to a diagonal matrix
Quote:

Originally Posted by EuropeanSon
A square matrix $\displaystyle A$ (of some size $\displaystyle n \times n$) satisfies the condition $\displaystyle A^2 - 8A +15I = 0.$ (a) Show that this matrix is similar to a diagonal matrix.

$\displaystyle p(\lambda)=\lambda^2-8\lambda+15=(\lambda-3)(\lambda-5)$ is an annihilator polynomial of $\displaystyle A$ . This implies that the minimum polynomial of $\displaystyle A$ is $\displaystyle \mu(\lambda)=\lambda-3$ or $\displaystyle \mu(\lambda)=\lambda-5$ or $\displaystyle \mu(\lambda)=(\lambda-3)(\lambda-5)$ . In all cases, $\displaystyle A$ is diagonalizable. Try (b).
• Aug 2nd 2011, 05:44 AM
EuropeanSon
Re: Show this matrix is similar to a diagonal matrix
Quote:

Originally Posted by FernandoRevilla
$\displaystyle p(\lambda)=\lambda^2-8\lambda+15=(\lambda-3)(\lambda-5)$ is an annihilator polynomial of $\displaystyle A$ . This implies that the minimum polynomial of $\displaystyle A$ is $\displaystyle \mu(\lambda)=\lambda-3$ or $\displaystyle \mu(\lambda)=\lambda-5$ or $\displaystyle \mu(\lambda)=(\lambda-3)(\lambda-5)$ . In all cases, $\displaystyle A$ is diagonalizable. Try (b).

Thanks. I have no idea, however, what an annihilator polynomial is, and hadn't come across the term before, and so I don't think this was a method which my lecturer intended me to use or would have accepted without proofs (this was an exam question in a paper I had a few months ago).
• Aug 2nd 2011, 06:14 AM
EuropeanSon
Re: Show this matrix is similar to a diagonal matrix
I looked over our course contents, and it appears we did have sufficient material covered to solve it in this manner. Thanks, I'll attempt part (b) now.
• Aug 2nd 2011, 07:17 AM
FernandoRevilla
Re: Show this matrix is similar to a diagonal matrix
Quote:

Originally Posted by EuropeanSon
I looked over our course contents, and it appears we did have sufficient material covered to solve it in this manner. Thanks, I'll attempt part (b) now.

All right, a little help for (b). If $\displaystyle k=3s+5r$ with $\displaystyle s\geq 0,r\geq 0$ integers, choose

$\displaystyle A=\textrm{diag}\;(\underbrace{3,\ldots,3}_{\mbox{s times}},\underbrace{5,\ldots,5}_{\mbox{r times}})$
• Aug 3rd 2011, 01:22 PM
EuropeanSon
Re: Show this matrix is similar to a diagonal matrix
Quote:

Originally Posted by FernandoRevilla
All right, a little help for (b). If $\displaystyle k=3s+5r$ with $\displaystyle s\geq 0,r\geq 0$ integers, choose

$\displaystyle A=\textrm{diag}\;(\underbrace{3,\ldots,3}_{\mbox{s times}},\underbrace{5,\ldots,5}_{\mbox{r times}})$

Ah, yes, in retrospect I'm shocked at not figuring that out. After that it was simple. I was messing around with all sorts of things before, but overlooked letting the $\displaystyle s$ or $\displaystyle r$ variables equal zero for some reason. Thanks!
• Aug 3rd 2011, 01:58 PM
FernandoRevilla
Re: Show this matrix is similar to a diagonal matrix
Don't worry!. Such things usually happen in Mathematics. :)