Two \(n \times n\) matrices \(A\) and \(B\) over a field \(F\) are said to be similar if there exists an \(n \times n\) invertible matrix \(T\) over \(F\) such that \(TAT^{-1} = B\). Exhibit three \(3 \times 3\) matrices over \(\mathbb{Q}\) no two of which are similar such that \(-2\) is the only rational eigenvalue of each of the matrices. For each, determine its elementary divisors, minimal polynomial and characteristic polynomial.
Solution
1. Let\[A_1 = \begin{pmatrix} -2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -2 \end{pmatrix}, \qquad A_2 = \begin{pmatrix} -2 & 1 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -2 \end{pmatrix}, \qquad A_3 = \begin{pmatrix} -2 & 1 & 0 \\ 0 & -2 & 1 \\ 0 & 0 & -2 \end{pmatrix}.\]Each has characteristic polynomial \((x+2)^3\), and \(-2\) is its only eigenvalue.
Each \(A_j\) is upper triangular with every diagonal entry \(-2\), so \(\det(xI-A_j)=(x+2)^3\). The eigenvalues are the roots of the characteristic polynomial, so \(-2\) is the only one.
2. The elementary divisors and minimal polynomials are:
- \(A_1\): elementary divisors \(x+2,\ x+2,\ x+2\); minimal polynomial \(x+2\).
- \(A_2\): elementary divisors \((x+2)^2,\ x+2\); minimal polynomial \((x+2)^2\).
- \(A_3\): elementary divisor \((x+2)^3\); minimal polynomial \((x+2)^3\).
The matrices are in Jordan form with eigenvalue \(-2\) and Jordan blocks of sizes \((1,1,1)\), \((2,1)\), and \((3)\). A Jordan block of size \(k\) with eigenvalue \(-2\) contributes the elementary divisor \((x+2)^k\). The characteristic polynomial is the product of the elementary divisors, and the minimal polynomial is their least common multiple, here \((x+2)^m\) for the largest block size \(m\) [@DF04].
3. No two of \(A_1,A_2,A_3\) are similar.
Similar matrices have the same elementary divisors [@DF04]. The three lists in step 2 are distinct.
4. Q.E.D.
Steps 1 and 3 show that \(A_1,A_2,A_3\) are pairwise nonsimilar \(3\times3\) matrices over \(\QQ\) whose only rational eigenvalue is \(-2\), and steps 1 and 2 give their characteristic polynomials, elementary divisors, and minimal polynomials.