symmetric and orthogonal matrices are connected?

Question

Are the set of all symmetric matrices and orthogonal matrcises connected in $M_n(\mathbb{R})$ and $M_n(\mathbb{C})$?completely out of idea.plz help

score 7 · Accepted Answer · answered Apr 26 '12 at 18:52

7

Consider the images of each set under the determinant map. Are either of the images connected?

answered Apr 26 '12 at 18:52

JSchlather

aha! in that case $O_n(\mathbb{R})$ is not connected but Symmetric matrices can have det value any real number so they are connected – Myshkin Apr 26 '12 at 19:00
4

Note that the preimage of a connected set under continuous function might be disconnected. So you can only argue a set being $not$-connected if its continuous image is $not$-connected. – T. Eskin Apr 26 '12 at 19:01
ok I am feeling that Symm matrices are path connected? – Myshkin Apr 26 '12 at 19:08
2

@Makuasi sure just show there's a path between every symmetric matrix and the 0 matrix. – JSchlather Apr 26 '12 at 19:28
1

@jacob, every vector space is path connected, so are syymm matrces.... – Myshkin Apr 27 '12 at 06:36

score 1 · Answer 2 · answered Sep 14 '20 at 15:19

1

Indeed the set of all real symmetric matrix is a convex set. So, obviously path connected

answered Sep 14 '20 at 15:19

SJA

2 Answers2