Results 1 to 2 of 2

Thread: Orthonormal Basis

  1. #1
    Member
    Joined
    Aug 2009
    Posts
    78

    Orthonormal Basis

    Let $\displaystyle B$ be a $\displaystyle nxn$ matrix and $\displaystyle V_{\lambda} =\{v \in \mathbb{C}^n : Bv=\lambda v\}$. Let $\displaystyle \{e_1^\lambda ,e_2^\lambda, ... ,e_{k(\lambda)}^\lambda\}$ be an orthonormal basis for $\displaystyle V_\lambda$ consisting of eigenvectors for $\displaystyle C_\lambda$ , where$\displaystyle C_\lambda$ is a self-adjoint linear transformation on $\displaystyle V_\lambda$.
    Prove that $\displaystyle \{e_i^\lambda : \lambda$ is an eigenvalue for $\displaystyle B$ and $\displaystyle 1\le i\le k(\lambda)\}$ is an orthonormal basis for $\displaystyle \mathbb{C}^n$.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,781
    Thanks
    3030
    Since we are given that these vectors are an orthonormal basis for $\displaystyle V_\lambda$ and we want to prove it is an orthonormal basis for $\displaystyle C^n$, it would appear that we just want to prove that $\displaystyle V_\lambda= C^n$. Self adjoint transformations have several nice properities that you would want to prove first, if you haven't already:
    All eigenvalues are real.
    If u and v are eigenvectors corresponding to distinct eigenvalues, then u and v are orthogonal.
    If $\displaystyle \{v_1,v_2, ..., v_i\}$ are eigenvectors and we restrict the transformation to the orthogonal complement of the span of $\displaystyle \{v_1,v_2, ..., v_i\}$ the restriction is still self adjoint.

    That way, we can start with one eigenvector, the restict ourselves to the orthogonal complement of its span, find a second eigenvector, etc. That way you should be able to show that there are, in fact, n independent eigenvectors and so $\displaystyle V_\lambda= C^n$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Orthonormal basis
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Apr 3rd 2011, 06:30 AM
  2. Orthonormal basis
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: Aug 30th 2010, 05:10 AM
  3. Orthonormal Basis
    Posted in the Advanced Algebra Forum
    Replies: 23
    Last Post: May 14th 2010, 07:14 AM
  4. Orthonormal basis
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Jun 12th 2009, 08:13 PM
  5. Orthonormal basis
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 25th 2009, 07:47 AM

Search Tags


/mathhelpforum @mathhelpforum