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Consider the set of all $n\times n$ matrices with real entries, considered as the space $\mathbb{R^{n^2}}$ What can we say about connectedness and compactness of the following sets?

  1. The set of all orthogonal matrices.
  2. The set of all matrices with trace equal to unity.
  3. The set of all symmetric and positive definite matrices.

I need help to understand basic concepts of solving these kind of problems. I mean how to show connectedness or compactness when we have set of matrices with certain specific properties. As mentioned above. Thanks

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    @copper.hat Definitely sir2012-05-15

1 Answers 1

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Some orthogonal matrices have determinant $+1$ and others $-1$. The determinant is a continuous function. Therefore the set of orthogonal matrices is not connected. It is compact, since it's closed and bounded in the set of all $n\times n$ matrices. It's closed because it's the inverse-image of a one-point set under the continuous function $M\mapsto MM^T$. For boundedness, consider the norm $\|M\|=\sup\{\|Mx\| : \|x\|=1, x\in\mathbb{R}^{n\times 1}\}$. With this norm, every orthogonal matrix has norm $1$. Show that the topology induced by that norm is the same as that induced by summing the squares of the entries, and you've got boundedness.

The diagonal matrix with entries $x,1-x,0,0,0,\ldots$ has trace $1$ and does not approach a limit as $x\to\infty$, so that set cannot be compact.

If $M$ is positive-definite, then so is $cM$ for positive scalars $c$, and as $c\to\infty$, there is no limit, so that set is not compact.

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    How is done sir?2012-05-14