# Showing a nonsingular matrix

1. Nov 18, 2009

### newreference

1. The problem statement, all variables and given/known data

$$S \in C^{mxm}$$ is skew-hermitian. $$S^{*}=-S$$
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)$$\neq$$0 since (I-A) is nonsingular.
Other attempt was to try to show $$(I-S)(I-S)^{-1}=(I-S)^{-1}(I-S)=I$$.

2. Nov 18, 2009

### Staff: Mentor

You can't start with det(I - S) $\neq$ 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) $\neq$ 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. Nov 18, 2009

### Dick

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?