Repeat Example CAEHW by choosing x =

Form the matrix C whose columns are x, Ax, A²x, A³x, A⁴x, A⁵x and row- reduce the matrix,

The simplest possible relation of linear dependence on the columns of C comes from using 

scalars α₄  1 and α₅ = α₆ = 0 for the free variables in a solution to LS(C, 0). 

The remainder of this solution is α₁ = 3, α₂ = −1, α₃ = −3. 

This solution gives rise to the polynomial

p(x) = 3 − x − 3x² + x³ = (x − 3)(x − 1)(x + 1)

which then has the property that p(A)x = 0.

No matter how you choose to order the factors of p(x), the value of k (in the language of

 Theorem EMHE and Example CAEHW) is k = 2. 

For each of the three possibilities, we list the resulting eigenvector and the associated eigenvalue:

Note that each of these eigenvectors can be simplified by an appropriate scalar multiple,

but we have shown here the actual vector obtained by the product specified in the theorem.