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

In summary: This can be used to find the minimal polynomial of ##T## with respect to each standard basis vector in part b). In part c), you can find the eigenvalues by finding the roots of the minimal polynomial and then finding the corresponding eigenvectors. The eigenspaces will be the spaces spanned by the eigenvectors corresponding to each eigenvalue. Finally, in part d), you can find the minimal polynomial of ##T## overall by taking the least common multiple of the minimal polynomials with respect to each standard basis vector.
  • #1
toni07
25
0

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.
 
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  • #2
toni07 said:

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##.
 

1. What is a T-cyclic subspace?

A T-cyclic subspace is a subspace of a vector space that is generated by a single vector, called a cyclic vector, under repeated application of a linear transformation T. This means that all other vectors in the subspace can be obtained by applying powers of T to the cyclic vector.

2. How do you find a T-cyclic subspace?

To find a T-cyclic subspace, you can start by choosing a vector in the vector space and calculating its powers under the linear transformation T. If the resulting set of vectors forms a basis for the vector space, then the subspace generated by this vector is T-cyclic. If not, you can repeat this process with a different vector until you find a basis for the T-cyclic subspace.

3. What is a minimal polynomial in relation to eigenvalues and eigenvectors?

The minimal polynomial of a linear transformation T is the monic polynomial of smallest degree that has T as a root. In other words, it is the polynomial that when evaluated at T, results in the zero transformation. The eigenvalues of T are the roots of the minimal polynomial, and the eigenvectors are the corresponding vectors in the null spaces of the powers of T.

4. How do you calculate the minimal polynomial of a linear transformation?

To calculate the minimal polynomial of a linear transformation, you can start by finding the characteristic polynomial, which is the determinant of the matrix representing the linear transformation. Then, you can use the Cayley-Hamilton theorem to show that the minimal polynomial divides the characteristic polynomial. Finally, you can use the properties of the minimal polynomial to determine the monic polynomial of smallest degree that has the linear transformation as a root.

5. How are eigenvalues and eigenvectors related to T-cyclic subspaces?

Eigenvalues and eigenvectors play a crucial role in determining T-cyclic subspaces. The eigenvalues are the roots of the minimal polynomial, which in turn determines the structure of the T-cyclic subspace. The eigenvectors are the basis vectors for the T-cyclic subspace, and their corresponding eigenvalues determine the powers of T that are applied to the cyclic vector to generate the other vectors in the subspace.

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