# Find T-cyclic subspace, minimal polynomials, eigenvalues, eigenvectors

1. Nov 25, 2013

### toni07

1. The problem statement, all variables and given/known data

Let T: R^6 -> R^6 be the linear operator defined by the following matrix(with respect to the standard basis of R^6):
(0 0 0 0 0 1
0 0 0 0 1 0
1 0 0 0 0 0
0 0 0 1 0 0
0 1 0 0 0 0
0 0 1 0 0 0 )
a) Find the T-cyclic subspace generated by each standard basis vector.
b) Find all the minimal polynomial of T with respect to each standard basis vector.
c) Find all eigenvalues, eigenvectors, and eigenspaces of T(over R).
d) Find the minimal polynomial of T.

2. Relevant equations

3. The attempt at a solution
I used the definition of T-cyclic subspace of span{x, T(X), T^2(X),...} for part a, since the basis is already in the matrix, but I don't think I'm right. I don't understand how I'm supposed to find minimal polynomial with respect to basis, I tried to find the determinant of the matrix, but I don't really think that's the right way to go since the matrix contains standard basis. Please help.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Nov 25, 2013

### Ray Vickson

If $e$ is any vector (in particular, a basis vector), look at the sequence of vectors $e_0 = e, \:e_1 = Te, \:e_2 = T^2 e, \ldots$. If $e_0,\ldots, e_{r-1}$ are linearly independent but $e_0, \:e_1, \ldots, e_r$ are linearly dependent, the minimial polynomial of $e$ has degree $r$, and the linear combination $e_r + c_{r-1} e_{r-1} + \ldots + c_0 e_0 = 0$ gives the minimal polynomial of $e$ as $x^r + c_{r-1} x^{r-1} + \ldots + c_0$.