Number of similarity classes with same characteristic equation.

Question

Let say I have a characteristic polynomial e.g. $(\lambda-1)^4(\lambda-2)^3$. How can I find the number of non-similar matrices with this characteristic equation?

Answer 1

Since your characteristic polynomial "splits" into linear factors,

$$ (\lambda - 1)^4 (\lambda - 2)^3 $$

we can find a Jordan normal form (over the complex field) or a "real" Jordan normal form (over the field of real numbers) to represent each possible class of similar matrices having this characteristic polynomial.

For each eigenvalue $\lambda =1,2$ the algebraic multiplicity can be partitioned in several ways. The multiplicity $4$ of the first eigenvalue can be expressed:

$$ 4 = 1+1+1+1 $$ $$ 4 = 1+1+2 $$ $$ 4 = 1+3 $$ $$ 4 = 2+2 $$ $$ 4 = 4 $$

In terms of the Jordan blocks, there could be four blocks of size $1$ (geometric multiplicity $4$), two blocks of size $1$ and one of size $2$, one block of size $1$ and one of size $3$, two blocks each of size $2$, or a single block of size $4$.

Similarly the second eigenvalue with algebraic multiplicity $3$ can be expressed in partitions:

$$ 3 = 1+1+1 $$ $$ 3 = 1+2 $$ $$ 3 = 3 $$

So we have five possibilities for the Jordan blocks of $\lambda = 1$ and three possibilities for the Jordan blocks of $\lambda = 2$. Altogether there are fifteen ($5\times 3$) possible similarity classes if the matrix is considered over the real or the complex numbers.