Results 1 to 2 of 2

Math Help - M 2x2 matrices of C with standard basis.

  1. #1
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    M 2x2 matrices of C with standard basis.

    T(A)=\begin{bmatrix}0&1\\1&0\end{bmatrix}\times A\times\begin{bmatrix}0&1\\1&0\end{bmatrix}

    The matrix of transformation is

    \begin{bmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0 \end{bmatrix}

    Eigenvalues are

    \lambda=1,1,-1,-1

    So finding the eigenspace.

    When \lambda=1,
    \begin{bmatrix}-1&0&0&1\\0&-1&1&0\\0&1&-1&0\\1&0&0&-1\end{bmatrix}\Rightarrow\begin{bmatrix}-1&0&0&1\\0&-1&1&0\\0&0&0&0\\0&0&0&0\end{bmatrix}

    So does this mean the Eigenspace is for lambda = 1 is

    \begin{bmatrix}1&0\\0&1\end{bmatrix}, \ \begin{bmatrix}0&1\\1&0\end{bmatrix}
    Last edited by Opalg; November 30th 2011 at 12:21 AM. Reason: fixed LaTeX
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7

    Re: M 2x2 matrices of C with standard basis.

    Quote Originally Posted by dwsmith View Post
    T(A)=\begin{bmatrix}0&1\\1&0\end{bmatrix}\times A\times\begin{bmatrix}0&1\\1&0\end{bmatrix}

    ... does this mean the Eigenspace is for lambda = 1 is

    \begin{bmatrix}1&0\\0&1\end{bmatrix}, \ \begin{bmatrix}0&1\\1&0\end{bmatrix}
    Yes it does (or rather, the eigenspace is the set of linear combinations of those two matrices). You can check that by taking A to be either of those matrices and verifying that T(A) = A.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: March 10th 2011, 01:39 PM
  2. standard basis
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 19th 2010, 07:12 AM
  3. Standard Matrices and basis vectors
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: June 23rd 2010, 08:28 PM
  4. Standard Basis of the complex filed?
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 16th 2009, 07:46 PM
  5. Basis Matrices
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 1st 2008, 12:22 PM

/mathhelpforum @mathhelpforum