Results 1 to 2 of 2

Thread: T is a projection and normal, then T is orthogonal projection

  1. #1
    Member Last_Singularity's Avatar
    Joined
    Dec 2008
    Posts
    157

    T is a projection and normal, then T is orthogonal projection

    Question: Let $\displaystyle T$ be a normal operator on a finite-dimensional inner product space. Prove that if $\displaystyle T$ is a projection, then $\displaystyle T$ is also an orthogonal projection.

    Attempt: Since $\displaystyle T$ is normal, $\displaystyle TT^* = T^*T$ and since $\displaystyle T$ is a projection, then $\displaystyle T=T^2$.

    Since $\displaystyle T$ is an orthogonal projection if and only if $\displaystyle T^2 = T = T^*$, we already have the left equality by that fact that $\displaystyle T$ is a projection. We only need to show that $\displaystyle T=T^*$ and we'll be done.

    Somehow, I figure that I can do this by showing that $\displaystyle <T(x),y>=<x,T(y)>$ but how?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    Since $\displaystyle T$ is normal, it is unitarily equivalent to a diagonal matrix. Since $\displaystyle T=T^2$, its eigenvalues can be only $\displaystyle \pm 1$. Thus $\displaystyle T=UDU^*$ and $\displaystyle T^*=U^{**}D^*U^*=UDU^*=T$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Orthogonal projection.
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Oct 24th 2011, 01:27 AM
  2. Orthogonal projection
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Apr 18th 2010, 05:58 AM
  3. Orthogonal Projection
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Feb 17th 2010, 06:06 AM
  4. Orthogonal projection
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Jun 24th 2009, 07:44 PM
  5. orthogonal projection
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Sep 6th 2008, 07:00 PM

Search Tags


/mathhelpforum @mathhelpforum