# Showing a nonsingular matrix

## Homework Statement

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

## 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 $$(I-S)(I-S)^{-1}=(I-S)^{-1}(I-S)=I$$.

Unfortunately, I could get nowhere. Thanks in advance for your help...

## Answers and Replies

Mark44
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.

Dick
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?