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Projection operator in spectrum theory

  1. Feb 10, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that if A is a normal operator in an n-dimensional vector space, and if A has r distinct eigenvalues a1,a2,...ar, then the projection operator onto the subspace with eigenvalue ai can be written as:

    Pi=[(A-a1)...(A-aa-1)...(A-ar)]/[(ai-a1) ...(ai-ai-1)...(ai-ai+1)...(a1-ar)]


    2. Relevant equations

    APiz=aiPiz

    A= (i=1 to r)[tex]\Sigma[/tex]aiPi


    3. The attempt at a solution
    First, sorry I'm a noob at text formatting.
    The equations I put in are basically my attempt at the solution. They weren't included in the problem, but I was looking around and thought that they are appropriate. I really don't know where to start with this, and after staring at it for a good hour or so, I've decided to ask for direction.
     
  2. jcsd
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