Showing a nonsingular matrix

  • #1

Homework Statement



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


Homework Equations





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...
 

Answers and Replies

  • #2
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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.
 
  • #3
Dick
Science Advisor
Homework Helper
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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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