# Minimal polynomial

1. Oct 12, 2013

### cristina89

1. The problem statement, all variables and given/known data
Given the matrix
2 0 0 0 0 0 0
1 2 0 0 0 0 0
0 1 2 0 0 0 0
0 0 1 2 0 0 0
0 0 0 0 2 0 0
0 0 0 0 1 2 0
0 0 0 0 0 0 2

What is the minimal polynomial?

2. Relevant equations

-

3. The attempt at a solution

This is the Jordan form, so I guess the solution is just m(t) = (t-2)7 but I don't know if it's right. Can anyone help me?

2. Oct 12, 2013

### I like Serena

Hi cristina89!

The minimal polynomial P of a square matrix A is the unique monic polynomial of least degree, m, such that P(A) = 0.

The degree of the minimal polynomial is determined by the size of the largest Jordan block, which is 4 in your case.
So the minimal polynomial is m(t) = (t-2)4.

Indeed $(A-2I)^4=0$.

3. Oct 12, 2013

### cristina89

Thank you so much!! :)

4. Oct 12, 2013

### Ray Vickson

The method suggested above is by far the simplest way to deal with this specific problem, but in a more general case you can use the algorithms employed by computer algebra systems, such as Maple: regard A, A^2, A^3,... as n^2-dimensional vectors, then find the smallest k such the vectors I, A, A^2,..,A^k are linearly dependent---essentially, using standard linear algebra methods. This will also deliver the coefficients and hence the minimal polynomial.