Compute the tangent space at the unit matrix

Question

Compute the tangent space $T_pM$ of the unit matrix $p=I$ when $$(i)\,M=SO(n)\\ (ii)\,M=GL(n)\\ (iii)\,M=SL(n).$$

My attempt: I think I have computed the tangent space in the case that $M=SL(n)$. We can write $SL(n)=\det^{-1}(1)$ and I've proved in an earlier exercise that $1$ is a regular value and that $D\det(I)\cdot H= \text{trace } H$. So the tangent space consists of precisely those matrices $H$ which have vanishing trace.

I don't know how to proceed in the other two cases. Can we write $SO(n)$ or $GL(n)$ as the pre-image of a regular value?

Answer 1

Basically you write down the defining equation of your group. I'll do it for $O(n)$. Something like this:

$$M^TM=1$$

The tangent space can be defined as the equivalence class of directions of curves in your manifold. Let $M_t\in O(n)$ with $M_0=1$ and $\frac{d}{dt}\bigr|_{t=0}M_t=X$. Then

$$ \frac{d}{dt}\bigr|_{t=0}M_t^TM_t=0 $$

hence $$ X^T+X=0 $$

So you have shown that the tangent directions are contained in the skew symmetric matrices. Now you also have to show that any skew symmetric matrix can occur as a tangent direction. This can be done using the matrix exponential. Given $X$ skew symmetric, study the path $M_t=\exp (tX)$ Proceed similarly for the other groups.

Answer 2

These are foundational calculations in what used to be called the theory of "continuous groups", now called Lie groups. The group structures are in fact smooth in today's terminology, not only continuous. The beginning material of any text on Lie groups and Lie algebras will have what you are looking for. The respective answers for $SO(n), GL(n)$ and $SL(n)$ are

(i) skew-symmetric matrices, (ii) all matrices, (iii) trace 0 matrices.