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Showing a nonsingular matrix

  1. Nov 18, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex]S \in C^{mxm}[/tex] is skew-hermitian. [tex]S^{*}=-S[/tex]
    I need to show that I-S is nonsingular.


    2. Relevant equations



    3. The attempt at a solution
    There are two things that I thought. First was start with det(I-A)[tex]\neq[/tex]0 since (I-A) is nonsingular.
    Other attempt was to try to show [tex](I-S)(I-S)^{-1}=(I-S)^{-1}(I-S)=I[/tex].

    Unfortunately, I could get nowhere. Thanks in advance for your help...
     
  2. jcsd
  3. Nov 18, 2009 #2

    Mark44

    Staff: Mentor

    You can't start with det(I - S) [itex]\neq[/itex] 0, since that's essentially what you need to prove. Also, by writing (I - S)(I - S)-1, you are assuming the existence of the inverse of I - S, which is equivalent to what you want to prove.

    I think your best bet is to show (not assume) that det(I - S) [itex]\neq[/itex] 0. Alternatively, you might assume that det(I - S) = 0 and see if you can arrive at a contradiction with your other assumption that S* = -S.
     
  4. Nov 18, 2009 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You want to show I-S has kernel {0}, hence show (I-S)x is nonzero if x is nonzero. Can you show the inner product <(I-S)x,(I-S)x> is nonzero?
     
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