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  1. The set of all upper triangular matrices in $\mathbb M(n,\mathbb R)$ such that all their eigenvalues satisfy $|\lambda| \leq 2$.

  2. The set of all real symmetric matrices in $\mathbb M(n,\mathbb R)$ such that all their eigenvalues satisfy $|\lambda| \leq 2$.

  3. The set of all diagonalisable matrices in $\mathbb M(n,\mathbb R)$ such that all their eigenvalues satisfy $|\lambda| \leq 2$.

Struggler
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    Did you already come to some conclusion yourself? Do you have any thoughts on what the result should be? My first hint is that for all the sets that are not compact, a $2\times2$ counterexample exists. – 5xum Jan 23 '14 at 13:43
  • i know all the sets are bounded, how to check closednes – Struggler Jan 23 '14 at 17:59
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    Double check your proof they are bounded. If you find nothing wrong with it, post it and we'll see where the problem is. – 5xum Jan 23 '14 at 18:36

1 Answers1

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$1$ is not bounded, consider the set of all matrices having diagonal entries $a_{11}, a_{22}$ as 1 and $a_{12}=n, a_{21}=0$. this set is unbounded as a subset of $M_2(\mathbb{R})$ hence not compact.

$3$ is not bounded, consider the set of all matrices having diagonal entries $a_{11}, a_{22}$ as 1 and 2 respectvely and $a_{12}=n, a_{21}=0$. this set is unbounded as a subset of $M_2(\mathbb{R})$ hence not compact.

Myshkin
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