Results 1 to 3 of 3

Math Help - Prove there is no eigenbasis

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    20

    Prove there is no eigenbasis

    Given an operator T that has in some basis matrix
    M(T)=
    <br />
\begin{bmatrix}<br />
1 & 1\\ <br />
0 & 1\\<br />
\end{bmatrix}<br />
    prove there is no eigenbasis for T.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5
    Quote Originally Posted by tjkubo View Post
    Given an operator T that has in some basis matrix
    M(T)=
    <br />
\begin{bmatrix}<br />
1 & 1\\ <br />
0 & 1\\<br />
\end{bmatrix}<br />
    prove there is no eigenbasis for T.
    <br />
\begin{bmatrix}<br />
1 & 1\\ <br />
0 & 1\\<br />
\end{bmatrix}<br />

    \lambda_1=\lambda_2=1

    <br />
\begin{bmatrix}<br />
0 & 1\\ <br />
0 & 0\\<br />
\end{bmatrix}\rightarrow <br />
x_1\begin{bmatrix}<br />
1\\ <br />
0\\<br />
\end{bmatrix}<br />

    This matrix isn't diagonalizable since the eigenspace is less n.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,237
    Thanks
    1795
    In other words, find the eigenvalues and all eigenvectors corresponding to those eigenvalues. If, for an n by n matrix, there are not n independent eigenvectors, they cannot form a basis for the n dimensional space and so there is no "eigenbasis".
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove a(AB)=(aA)B=A(aB) ..
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 29th 2010, 05:14 AM
  2. how to prove abs(a-b) <= abs(a) + abs(b)?
    Posted in the Calculus Forum
    Replies: 5
    Last Post: September 27th 2010, 07:52 PM
  3. Prove: f is one-to-one iff f is onto
    Posted in the Discrete Math Forum
    Replies: 12
    Last Post: June 25th 2010, 11:02 AM
  4. Replies: 2
    Last Post: August 28th 2009, 03:59 AM
  5. sum prove
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: February 18th 2009, 02:01 AM

Search Tags


/mathhelpforum @mathhelpforum