Results 1 to 2 of 2

Thread: Spectral decomposition and invertibility

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

    Spectral decomposition and invertibility

    Question: Let $\displaystyle T$ be a normal operator on a finite-dimensional complex inner product space $\displaystyle V$. Consider spectral decomposition $\displaystyle T = \lambda_1 T_1 + ... + \lambda_k T_k$ and prove that $\displaystyle T$ is invertible if and only if $\displaystyle \lambda_i \neq 0$ for $\displaystyle i \leq i \leq k$.

    Any pointers would be helpful - I am unable to begin currently.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    3
    Quote Originally Posted by Last_Singularity View Post
    Question: Let $\displaystyle T$ be a normal operator on a finite-dimensional complex inner product space $\displaystyle V$. Consider spectral decomposition $\displaystyle T = \lambda_1 T_1 + ... + \lambda_k T_k$ and prove that $\displaystyle T$ is invertible if and only if $\displaystyle \lambda_i \neq 0$ for $\displaystyle i \leq i \leq k$.

    Any pointers would be helpful - I am unable to begin currently.

    Well, this is ALWAYS true: any operator (of a finite-dimensional vector space $\displaystyle V$) over any field is invertible iff all its eigenvalues are different from zero, and the proof is painfully simple: $\displaystyle T$ is NOT invertible iff $\displaystyle Ker(T)\ne {0}\,\Longleftrightarrow\,\exists\,0\ne v\in V\,\,\,s.t.\,\,Tv=0=0\cdot v\,\Longleftrightarrow\,0$ is an eigenvaue of $\displaystyle T$.

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Spectral Decomposition and Polynomials
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Nov 17th 2011, 12:39 PM
  2. spectral decomposition
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Jun 15th 2011, 08:53 AM
  3. Spectral Decomposition of a Matrix
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Apr 17th 2010, 11:03 AM
  4. wold decomposition and spectral density
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: May 25th 2009, 01:02 PM
  5. Jordan Decomposition to Schur Decomposition
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Mar 30th 2009, 01:52 PM

Search Tags


/mathhelpforum @mathhelpforum