Show this matrix is similar to a diagonal matrix

A square matrix (of some size ) satisfies the condition

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

(b) Show that for every positive integer there exists a matrix satisfying the above condition with .

I have little or no idea where to go with this problem. I know that for (a), is obviously a diagonal matrix , but am not sure if this is of any use. I also am not sure if I can simply say , so , in which case, either (i) or , or (ii) with and , 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.

Re: Show this matrix is similar to a diagonal matrix

Re: Show this matrix is similar to a diagonal matrix

Quote:

Originally Posted by

**FernandoRevilla** is an annihilator polynomial of

. This implies that the minimum polynomial of

is

or

or

. In

**all** cases,

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).

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.

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 with integers, choose

Re: Show this matrix is similar to a diagonal matrix

Quote:

Originally Posted by

**FernandoRevilla** All right, a little help for (b). If

with

integers, choose

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 or variables equal zero for some reason. Thanks!

Re: Show this matrix is similar to a diagonal matrix

Don't worry!. Such things usually happen in Mathematics. :)