Being morse function for a determinant map on M(n)

Question

Show that the determinant map on M(n) is Morse function if n=2.

I know that f to be a morse function, all critical points for f must be nondegenerate.

But i dont know how i calculate the derivative of a determinant map

score 1 · Answer 1 · answered Apr 06 '16 at 17:12

1

Well, you do have $n=2$, after all. So write the matrix as $\begin{bmatrix} x&y\\z&w\end{bmatrix}$, and then you have no problem differentiating the determinant function. :)

answered Apr 06 '16 at 17:12

Ted Shifrin

115,160

What about n> 2 ? – diffgeo Apr 06 '16 at 17:14
I want to learn the general derivative of a determinant map for arbitrary n – diffgeo Apr 06 '16 at 17:15
1

Use the cofactor expansion of the determinant in general to compute the derivative with respect to $x_{ij}$. – Ted Shifrin Apr 06 '16 at 17:15

score 0 · Answer 2 · answered Nov 08 '19 at 14:20

0

Consider the diffeomorphism $M_n(\mathbb{R}) \stackrel{\sim}{=} \mathbb{R}^{n^2}$.

answered Nov 08 '19 at 14:20

Ryze

1,136

Being morse function for a determinant map on M(n)

2 Answers2