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Problem: If $S$ is a skew-symmetric matrix, show that $(I+S)(I-S)^{-1}$ is orthogonal.

This appeared on a list of standard questions asked of Princeton graduate students. It has been a while since I've studied linear algebra, and frankly I cannot even see why $(I-S)$ must be invertible.

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up vote 5 down vote accepted

I will assume we are in $M_n(\mathbb{R})$, since you want your matrix to be orthogonal.

The spectrum of a skew-symmetric matrix is contained in $i\mathbb{R}$, so the spectrum of $A=I-S$ does not contain $0$ and $A$ is invertible, in $M_n(\mathbb{C})$ first, hence in $M_n(\mathbb{R})$.

The matrix we are interested in is $A^*A^{-1}$. Note that $A$ and $A^*$ commute.

Now $$ A^*A^{-1}(A^*A^{-1})^*=A^*AA^{-1}(A^{-1})^*=A^*AA^{-1}(A^*)^{-1}=A^*A(A^*A)^{-1}=I. $$

So $A^*A^{-1}$ is indeed orthogonal.

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+1 You could also say that the real spectrum of a skew-symmetric matrix $S$ has $\mathrm{Sp}_{\Bbb R}(S)\subset\lbrace 0\rbrace$, so $A$ is invertible in $M_n(\Bbb R)$. – Olivier Bégassat Feb 4 '13 at 0:47

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