Let A be an nxn matrix such that (I+A) is invertible. I need to prove that the Cayley Transform of A, denoted by $A^c$, is such that $(I+A^c)$ is invertible. The Cayley Transform is defined as follows: $$A^c:=(I-A)(I+A)^{-1}$$ I tried showing that $det(I+A^c)\neq0$, but $(I+A^c)=(I+(I-A)(I+A)^{-1})$ and I don't know what formulas I can use for the addition of matrices under the determinant. I also tried explicitly calculating $(I+A^c)^{-1}=(I+(I-A)(I+A)^{-1})^{-1}$, but I got stuck as well. What equalities (or inequalities) can I use in any of these two cases, where I can use the fact that (I+A) is invertible?
2026-04-07 00:26:52.1775521612
Cayley transform a matrix that is invertible when added to the identity
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Just note that $(I+A^c)(I+A)=2I$.