MHB 25.1 find all possible Jordan Normal Forms of A

[karush]

nmh{1000}
Suppose that A is a matrix whose characteristic polynomial is
$$(\lambda-2)^2(\lambda+1)^2$$
find all possible Jordan Normal Forms of A (up to permutation of the Jordan blocks).ok i have been looking at examples so pretty fuzzy on this
for the roots are 2 and -1so my first stab at this is

$\left[\begin{array}{c} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 &2 \end{array}\right]$

 

[HOI]

When a matrix has repeating eigenvalues, the various Jordan forms will have "blocks" with those eigenvalues on the main diagonal and either "0" or "1" above them, depending on what the corresponding eigenvector are. Yes, the diagonal matrix with only "0" above the eigenvalues is a Jordan matrix where there are 4 independent eigenvectors (a "complete set" of eigenvectors or a basis for the space consisting of eigenvectors). In this case, both eigenvalues have "algebraic multiplicity" and "geometric multiplicity" 2. Other possible Jordan forms, are

$\begin{bmatrix}-1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{bmatrix}$
when there are two independent eigenvectors corresponding to eigenvalue 2 but only one (and multiples) corresponding to eigenvalue -1. We say that -1 and 2 both have "algebraic multiplicity" 2 and that 2 has "geometric multiplicity" 2 but that -1 has "geometric multiplicity 1".

$\begin{bmatrix}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2 \end{bmatrix}$
when there are two independent eigenvectors corresponding to eigenvalue -1 but only one (and multiples) corresponding to eigenvalue 2.We say that -1 and 2 both have "algebraic multiplicity" 2 and that -1 has "geometric multiplicity" 2 but that 2 has "geometric multiplicity 1".


and
$\begin{bmatrix}-1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2 \end{bmatrix}$ when there are only one eigenvector (and multiples) corresponding to each of eigenvalue -1 and eigenvalue 2.We say that -1 and 2 both have "algebraic multiplicity" 2 but that both -1 and 2 have "geometric multiplicity 1".

(An eigenvalue, $\lambda_0$, has "algebraic multiplicity" n if $\lambda- \lambda_0$ occurs to the nth power as a factor of the characteristic equation. It has "geometric multiplicity" n if the subspace of all eigenvector has dimension n. Of course the "geometric multiplicity" of a given eigenvalue is always less than or equal to its "algebraic multiplicity".)