Use the result of Exercise 11 and the Jordan form to prove the following. Let \(A\) and \(B\) be \(n \times n\) matrices over the field \(F\) which have the same characteristic polynomial \[ f = (x - c_1)^{d_1} \cdots (x - c_k)^{d_k} \] and the same minimal polynomial. Suppose also that for each \(i\) the solution spaces of \((A - c_iI)\) and \((B - c_iI)\) have the same dimension. If no \(d_i\) is greater than 6, then \(A\) and \(B\) are similar.
Solution
1. \(A\) and \(B\) are similar iff they have the same Jordan form (over the algebraic closure of \(F\)).
similarity is determined by the Jordan canonical form.
2. The Jordan form is determined by the sizes of the Jordan blocks for each eigenvalue \(c_i\).
the Jordan form is a direct sum of Jordan blocks.
3. For each eigenvalue \(c_i\), the number of Jordan blocks of size \(\ge j\) is \(\operatorname{rank}(A - c_i I)^{j-1} - \operatorname{rank}(A - c_i I)^j\).
standard formula for the Jordan block structure.
4. The ranks \(\operatorname{rank}(A - c_i I)^j\) are determined by the characteristic polynomial, the minimal polynomial, and the dimension of the eigenspace \(\ker(A - c_i I)\).
the characteristic polynomial gives \(d_i\) (the total size), the minimal polynomial gives the size of the largest block \(r_i\), and the eigenspace dimension gives the number of blocks.
5. Since \(A\) and \(B\) have the same characteristic polynomial, minimal polynomial, and eigenspace dimensions, they have the same ranks \(\operatorname{rank}(A - c_i I)^j\) for all \(j\).
4 (these three pieces of data determine the block structure).
6. The condition \(d_i \le 6\) ensures that the block structure is uniquely determined by these data (for \(d_i \le 6\), the partition of \(d_i\) into block sizes is determined by the largest block size \(r_i\) and the number of blocks).
5 and the hypothesis \(d_i \le 6\) (this is the content of Exercise 11, which shows the data determine the partition when \(d_i \le 6\)).
7. Hence \(A\) and \(B\) have the same Jordan form, so they are similar.
1, 3, 5, 6.
8. Q.E.D.
7.