Possible pairs of minimum polynomials and characteristic polynomials associated to a linear transformation

Question

I am trying to tackle the following problem:

Suppose that $V$ is a vector space of dimension $7$ over the field of real numbers, and $T$ is a linear transformation on $V$ which satisfies $T^4 = I$. Compute the possible minimum polynomials of $T$, and the characteristic polynomial that goes with each choice.

The problem is confusing because it suggests that there is exactly one characteristic polynomial associated to a possible choice for minimum polynomial, but I believe this is not the case for this particular linear transformation. Clearly, the minimal polynomial divides $x^4 - 1$; so it could be $x^4 - 1$ (the dimension of $V$ is $7$) which factored out as $(x^2 + 1)(x + 1)(x - 1)$. From here it is not hard to see that there are several characteristic polynomials that goes with this choice of minimum polynomial. Am I right or am I missing something here?

Answer 1

You are correct in your assertion that for some of the possible cases for the minimal polynomial of $T$, the characteristic polynomial of $T$ is not uniquely determined, however in each such case, the characteristic polynomial of $T$ is constrained to a specific form.

Consider each case separately . . .

• If the minimal polynomial of $T$ is $x-1$, then $T=I$, so the characteristic polynomial is $(x-1)^7$.$\\[8pt]$
• If the minimal polynomial of $T$ is $x+1$, then $T=-I$, so the characteristic polynomial is $(x+1)^7$.$\\[8pt]$
• If the minimal polynomial of $T$ is $x^2-1$, then the characteristic polynomial is not unique, but has the unique form $$(x+1)^k(x-1)^{7-k}$$ for some integer $k$ with $1\le k\le 6$.$\\[8pt]$
• Since the characteristic polynomial of $T$ has degree $7$, it must have a real root, hence the minimal polynomial can't be $x^2+1$.$\\[8pt]$
• If the minimal polynomial of $T$ is $(x^2+1)(x-1)$, then the characteristic polynomial is not unique, but has the unique form $$(x^2+1)^k(x-1)^{7-2k}$$ for some integer $k$ with $1\le k\le 3$.$\\[8pt]$
• If the minimal polynomial of $T$ is $(x^2+1)(x+1)$, then the characteristic polynomial is not unique, but has the unique form $$(x^2+1)^k(x+1)^{7-2k}$$ for some integer $k$ with $1\le k\le 3$.$\\[8pt]$
• If the minimal polynomial of $T$ is $(x^2+1)(x+1)(x-1)$, then the characteristic polynomial is not unique, but has the unique form $$(x^2+1)^j(x+1)^k(x-1)^{7-(2j+k)}$$ for some integers $j,k$ with $1\le j\le 2$ and $1\le k\le 6-2j$.

This completes the case analysis.

Based on the above results, I suspect that the intended wording of the problem was something like:

Suppose $V$ is a vector space of dimension $7$ over the field of real numbers, and $T$ is a linear transformation on $V$ which satisfies $T^4 = I$. Find all possibilities for the minimal polynomial of $T$, and for each case, determine the possible forms for the characteristic polynomial.