Linear Algebra question

EuropeanSon

Fremen

I need to think about this more, but maybe the Cayley-Hamilton theorem is relevant here. Did you cover that in class?

EuropeanSon

Fremen wrote: »

I need to think about this more, but maybe the Cayley-Hamilton theorem is relevant here. Did you cover that in class?

Apparently we did, but, I think, so briefly that I must have missed it. I'll see where that leads.

Edit: Thanks! It leads in useful directions.

EuropeanSon

A partial solution, then. The minimal polynomial of A must be (i) A-5I=0, (ii) A-3I=0, or (iii) (A-5I)(A-3I)=0. In all cases the matrix is diagonalizable, as (from wiki) an endomorphism is diagonalizable iff every Jordan block has size 1, which is equivalent to each root of the minimal polynomial having multiplicity 1 (and the minimal polynomial factors completely)

That's (a) done. Still thinking on (b)

equivariant

EuropeanSon wrote: »

A partial solution, then. The minimal polynomial of A must be (i) A-5I=0, (ii) A-3I=0, or (iii) (A-5I)(A-3I)=0. In all cases the matrix is diagonalizable, as (from wiki) an endomorphism is diagonalizable iff every Jordan block has size 1, which is equivalent to each root of the minimal polynomial having multiplicity 1 (and the minimal polynomial factors completely)

That's (a) done. Still thinking on (b)

For (b) just consider diagonal matrices that only have 3s or 5s on the diagonal. First, prove that all such matrices satisfy the given equation. Now you just need to show that any positive integer >= 8 can be written in the form 3m+5n where m and n are nonnegative integers.

PS Surely in an Irish mathematics forum, it should be referred to as the Hamilton-Cayley Theorem (that is how it was labelled when I studied linear algebra)

Fremen

Off topic, but a Chinese guy I work with didn't know what Pythagoras' theorem was. I drew the standard picture, and he goes "Ooooh! Gogu's theorem!"

equivariant

It just dawned on me that this is the old problem of using 3 litre and 5 litre bottles to measure out any number of litres (I like to think of that as the Willis-Jackson Theorem). Not often that Bruce Willis and WR Hamilton get featured in the same thread.

EuropeanSon

equivariant wrote: »

For (b) just consider diagonal matrices that only have 3s or 5s on the diagonal. First, prove that all such matrices satisfy the given equation. Now you just need to show that any positive integer >= 8 can be written in the form 3m+5n where m and n are nonnegative integers.

PS Surely in an Irish mathematics forum, it should be referred to as the Hamilton-Cayley Theorem (that is how it was labelled when I studied linear algebra)

I got that a few days ago, thanks! Forgot to post an update here.