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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

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

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    aha! in that case $O_n(\mathbb{R})$ is not connected but Symmetric matrices can have det value any real number so they are connected2012-04-26
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    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.2012-04-26
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    ok I am feeling that Symm matrices are path connected?2012-04-26
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    @Makuasi sure just show there's a path between every symmetric matrix and the 0 matrix.2012-04-26
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    @jacob, every vector space is path connected, so are syymm matrces....2012-04-27