How do you determine all possible Jordan Canonical forms of a matrix given its eigenvalues and their algebraic multiplicities

Question

This is what I’m given.

(1) Suppose A is a 5 × 5 matrix with eigenvalues -2 and 7 with corresponding algebraic multiplicities 2 and 3. Find all of the possible Jordan Canonical forms for A (up to rearrangement of blocks).

(2) Suppose we also know $ν(A + 2I) = ν(A − 7I) = 2$ and $ν((A − 7I)^2) = 3$. List all possible Jordan Canonical forms of A (again, up to rearrangement of blocks)

I’m not really sure where to start. I know the rank nullity theorem might apply so for -2, $$rk = 5 - 2 = 3$$ And for 7, $$rk = 5 - 2 = 3$$ For number one I think there should be 2 2x2 blocks, -2 on the diagonal of each and then a 1 or a 0 above the second -2. And then I think there should be 3 3x3 blocks for 7 each one having sevens on the diagonal but then one of them having 1 1 on the super diagonal, one having 2 1’s on the super diagonal and 1 having 0 ones.

For question 2 the only one that I think would work is the matrix that has 2 -2’s on the diagonal followed by 3 7’s.

I can’t figure out if I’m even on the right track and everything I look at confuses me with notation.

Answer 1

I will use the following notation:

• $J_n(\lambda)$ denotes an $n \times n$ Jordan block associated with the eigenvalue $\lambda$
• $\oplus$ denotes a diagonal sum, so that $A_1 \oplus \cdots \oplus A_n$ is the block-diagonal matrix $$ A_1 \oplus \cdots \oplus A_n = \pmatrix{A_1\\ & \ddots \\ && A_n} $$

For the first problem, your answer is correct. The key is to know that the algebraic multiplicity of an eigenvalue is the sum of the sizes of the Jordan blocks associated with that eigenvalue. So, because $-2$ has algebraic multiplicity $2$, the Jordan form either has the single block $J_2(-2)$ or the two blocks $J_1(-2),J_1(-2)$. Because $-7$ has algebraic multiplicity $3$, the Jordan form can have the single block $J_3(7),$ the two blocks $J_2(7),J_1(7)$, or the three blocks $J_1(7),J_1(7),J_1(7)$.

All together, that amounts two 6 distinct possibilities for the Jordan forms.

For question 2, there are two things at play here. First, note that $\nu(A - \lambda I)$ is the dimension of the eigenspace of $A$ associated with $\lambda$, i.e. the geometric multiplicity of the eigenvalue $\lambda$, which is equal to the total number of Jordan blocks assocaited with $\lambda$.

Second, $\nu((A - \lambda I)^2) - \nu(A - \lambda I)$ counts the number of Jordan blocks of $A$ associated with $\lambda$ that have size at least $2$.

With all that, we know that the Jordan form must contain 2 Jordan blocks in total for each eigenvalue, and that there is at least $3 - 2 = 1$ Jordan block for $7$ with size at least $2$. Assuming that the conditions of (2) are meant to be considered in addition to those from (1), this information that we get from $\nu((A - 7I)^2)$ is actually redundant: from the total number of Jordan blocks alone, we can deduce that there is only one possible Jordan form, namely $$ J = J_1(-2) \oplus J_{1}(-2) \oplus J_{2}(7) \oplus J_1(7). $$