Find Minimal Polynomial for Matrix: Solution Help

[cristina89]

Homework Statement

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?

Homework Equations

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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?

 

[I like Serena]

Homework Statement

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?

Homework Equations

-

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?

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

 

[cristina89]

Thank you so much! :)

 

[Ray Vickson]

Thank you so much! :)

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.