A complex matrix has characteristic polynomial x³(x − 1)⁴. How many similarity classes of such matrices are there?

Step-by-step solution

1. 1

   Use the fact that over $\mathbb{C}$ the Jordan form is a complete invariant. Two complex matrices are similar if and only if they have the same Jordan canonical form up to the order of the blocks. So counting similarity classes means counting possible Jordan forms — no other data can distinguish them.

2. 2

   Read the eigenvalues and their algebraic multiplicities off the characteristic polynomial. From $x^3(x-1)^4$ the matrix is $7\times 7$ with

   $\lambda=0 \text{ of multiplicity } 3,\qquad \lambda=1 \text{ of multiplicity } 4$

   The characteristic polynomial fixes the total size of the blocks at each eigenvalue, but not how that total is split into blocks — and that is exactly the remaining freedom.

3. 3

   Translate 'block sizes' into integer partitions. The Jordan blocks for one eigenvalue have sizes that are positive integers summing to its algebraic multiplicity, and their order does not matter. So the choices at eigenvalue $\lambda$ correspond one-to-one with the partitions of its multiplicity, counted by the partition function $p$.

4. 4

   List the partitions of $3$ and of $4$. For $\lambda=0$:

   $3,\quad 2+1,\quad 1+1+1\qquad\Longrightarrow\quad p(3)=3$

   For $\lambda=1$:

   $4,\quad 3+1,\quad 2+2,\quad 2+1+1,\quad 1+1+1+1\qquad\Longrightarrow\quad p(4)=5$

5. 5

   Multiply, because the two eigenvalues are chosen independently. The Jordan structure at $0$ places no constraint on the structure at $1$, so the total is the product:

   $p(3)\cdot p(4)=3\cdot 5=15$

   A representative of one class is $J_2(0)\oplus J_1(0)\oplus J_3(1)\oplus J_1(1)$; varying the two partitions independently sweeps out all $15$.

6. 6

   Sanity-check the extremes. The all-$1$ partitions on both sides give the diagonalisable matrix $\operatorname{diag}(0,0,0,1,1,1,1)$ — exactly one class. The single-block partitions give $J_3(0)\oplus J_4(1)$, the class with minimal polynomial $x^3(x-1)^4$ equal to the characteristic polynomial. Both are counted once among the $15$.