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

[toni07]

Homework Statement

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.

Homework Equations

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.

 

[Ray Vickson]

Homework Statement

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.

Homework Equations

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.

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 $x^r + c_{r-1} x^{r-1} + \ldots + c_0$.