I am not sure how to find the dimension of U(n).
Suggestions i have been given are: ((n^2) - n)) / 2
or: 2(n^2)
But i don't know which (if either) of these is correct, and how they are found.
Thanks for any help
I am not sure how to find the dimension of U(n).
Suggestions i have been given are: ((n^2) - n)) / 2
or: 2(n^2)
But i don't know which (if either) of these is correct, and how they are found.
Thanks for any help
Yes, in fact I mean "tangent space at the identity" in my post...
To get the tangent space at $\displaystyle A$, differentiate at $\displaystyle A$ the equation defining implicitly the manifold (here, $\displaystyle A^TA=I$) : it is the space of matrices $\displaystyle H$ such that $\displaystyle A^T H+H^T A=0$. If $\displaystyle A=I$, this reduces to $\displaystyle H=-H^T$. (More formally, the tangent space at A is the kernel of the differential at A of a (submersive?) map $\displaystyle \phi$ such that the manifold is a level set of $\displaystyle \phi$ (at least locally at $\displaystyle A$))