Results 1 to 2 of 2

Math Help - Change basis to get Jordan canonical form

  1. #1
    Member
    Joined
    Aug 2008
    Posts
    88

    Change basis to get Jordan canonical form

    In some basis e_1,e_2,...,e_n the matrix of an operator A is an n x n lower triangular matrix with diagonal entries \lambda, subdiagonal entries 1, and 0 for all other entries. In what basis does it have Jordan canonical form?


    It seems like you could just reverse the order of the basis (ie, e_n,e_{n-1},...,e_1). Is it enough to use the fact that A^T is in the required form to prove that this is the correct basis?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    44
    Quote Originally Posted by davesface View Post
    It seems like you could just reverse the order of the basis (ie, e_n,e_{n-1},...,e_1). Is it enough to use the fact that A^T is in the required form to prove that this is the correct basis?
    For example,

    B=(e_1,e_2,e_3),\quad B^*=(e_3,e_2,e_1)

    then,

    \begin{Bmatrix}Ae_1=\lambda e_1+e_2\\Ae_2=\lambda e_2+e_3\\Ae_3=\lambda e_3\end{matrix}\Rightarrow [A]_B=\begin{bmatrix}{\lambda}&{0}&{0}\\{1}&{\lambda}  &{0}\\{0}&{1}&{\lambda}\end{bmatrix}

    \begin{Bmatrix}Ae_3=\lambda e_3\\Ae_2=e_3+\lambda e_2\\Ae_1=e_2+\lambda e_1\end{matrix}\Rightarrow [A]_{B^*}=\begin{bmatrix}{\lambda}&{1}&{0}\\{0}&{\lam  bda}&{1}\\{0}&{0}&{\lambda}\end{bmatrix}

    It is easy to generalize.

    Regards.

    Fernando Revilla
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] 6x6 Jordan Canonical Form
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 30th 2011, 08:56 PM
  2. [SOLVED] Jordan Canonical Form
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 29th 2011, 10:43 PM
  3. jordan canonical form
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: May 26th 2010, 10:45 AM
  4. Jordan Canonical Form
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: November 4th 2008, 01:26 AM
  5. Jordan Canonical form question.
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: May 5th 2008, 10:43 AM

/mathhelpforum @mathhelpforum