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Let $S$ be the set of all symmetric positive definite matrices of size $n\times n$. Which of the following statements are true?

(a) $S$ is closed in $\mathbb{M}_n(\mathbb{R})$.
(b) $S$ is connected in $\mathbb{M}_n(\mathbb{R})$.
(c) $S$ is compact in $\mathbb{M}_n(\mathbb{R})$.

Only the option (a) & (b) are right. I guess that it is not bounded so (c) is not true. Am I correct?

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Can you prove it is not bounded? Do a particular case with $\,n=2\,$ and check you can easily construct a symmetric pos. def. matrix with as high a norm as wanted (I'm guessing you're taking the euclidean one in $\,\Bbb M_n(\Bbb R)\leq \Bbb R^{n^2}\,$ ...) –  DonAntonio Jan 19 '13 at 17:39
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1 Answer 1

It is just connected infact it is path connected. for $A,B$ such matrices we have $x^TAx\ge 0$, $X^TBx\ge 0$ so for $\lambda \in [0,1]$ we get $x^T[\lambda A+(1-\lambda)B]X\ge 0$.

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can you confirm me that is it closed or not? i think it is closed not fully sure. –  etuku Jan 19 '13 at 17:48
    
It is the inverse image of $0$ by the map $f(A) \mapsto (a_{i,j} - a_{j,i})_{i\neq j}$ which is clearly continuous. –  Alexander Thumm Jan 19 '13 at 17:50
    
thanks.then it is closed. –  etuku Jan 19 '13 at 17:50
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I must have somehow read only half of the question. The set of positive definite symmetric Matrices is not closed. To see this, $\lambda I$ is in $S$ for $\lambda > 0$ but not for $\lambda = 0$. –  Alexander Thumm Jan 19 '13 at 17:54
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