Minimum polynomial and canonical form

[squaremeplz]

Homework Statement

Hi all.

I have no clue on how to do this problem because I missed the class where he covered this so could someone please walk me through it.

A = [2 2 -5; 2 7 2; -5 -15 -4] where ; means new column

p(x) = (x-3)(x-1)^2

1) what are the choices of m(x)
2) find the minimum polynomial
3) find the jordan canonical form
d) find P s.t. P^(-1)AP

Homework Equations

The Attempt at a Solution

1)m(x) = (x-3)(x-1)^2

and m(x) = (x-3)(x-1)

2) no clue.. m(x) = (x-3)(x-1)?

do i have to plug in A for x or something?

3) no clue

4) no clue


I'm really sorry for this half a**ed attempt but I really do not have any material on this stuff. I tried researching the minimum polynomial problem but none of the examples are similar to mine. I just need a push in the right direction please


Thanks!

 

[dx]

Hi squaremeplease,

You found that the possible minimal polynomials are m₁(x) = (x-3)(x-1)² and m₂(x) = (x-3)(x-1).

m₂ has the smaller degree, so if m₂(A) = 0, then m₂ is the minimal polynomial. Otherwise, m₁ is the minimal polynomial. So all you have to do is check if (A - 3)(A - 1) = 0.

(Do you know the definition of minimal polynomial?)