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
symmetric and orthogonal matrices are connected?
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general-topology
1 Answers
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Consider the images of each set under the determinant map. Are either of the images connected?
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0aha! in that case $O_n(\mathbb{R})$ is not connected but Symmetric matrices can have det value any real number so they are connected – 2012-04-26
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2Note 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. – 2012-04-26
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0ok I am feeling that Symm matrices are path connected? – 2012-04-26
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2@Makuasi sure just show there's a path between every symmetric matrix and the 0 matrix. – 2012-04-26
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1@jacob, every vector space is path connected, so are syymm matrces.... – 2012-04-27