Difficulty finding minimal polynomial…

linear algebramatricesminimal-polynomials

I would like to find the minimal polynomial of the following matrix.

$$ A = \begin{pmatrix}
1& 2& 0& 0 \\
2& 1& 1& 1 \\
0& 0& 1& 2 \\
0& 0& 0& 1
\end{pmatrix}. $$

The characteristic polynomial of $A$ is
$$ c_A(x) = x^4 – 4x^3 + 2x^2 + 4x – 3 = (x-3)(x-1)^2(x+1). $$
Thus, the minimal polynomial must be either $c_A(x)$ or $(x-3)(x-1)(x+1)$.

Yet, neither $(A – 3I)(A-I)(A+I)$ nor $(A – 3I)(A-I)^2(A+I)$ yields the zero matrix (I double checked this in Mathematica).

I can't figure out what I've done wrong here…

EDIT:

Indeed, I simply made a computational error (though I can't quite figure out exactly where…)

Best Answer

Check your computation of matrix product, it works for me. 

octave:1> A = [ 1, 2, 0 , 0; 2, 1, 1, 1; 0, 0, 1, 2; 0, 0, 0, 1]
A =

   1   2   0   0
   2   1   1   1
   0   0   1   2
   0   0   0   1
octave:2> (A - 3* eye(4)) * (A -eye(4))*(A+eye(4))
ans =

   0   0   0   4
   0   0   0   0
   0   0   0  -8
   0   0   0   0

octave:3> (A - 3* eye(4)) * (A -eye(4))*(A-eye(4))*(A+eye(4))
ans =

   0   0   0   0
   0   0   0   0
   0   0   0   0
   0   0   0   0

The minimal polynomial is the characteristic polynomial.

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