Is the set of all matrices in M(n;R) all of whose eigenvalues satisfy the condition |λ|≤2. compact?

Is the set of all matrices in $M(n; R)$ all of whose eigenvalues satisfy the condition $|λ| ≤ 2.$ compact?

• Hint: take an upper triangular matrix with distinct eigenvalues less than 2, and let one of the off diagonal entries go to infinity. Jan 15, 2014 at 1:50

No. Using the standard topology on $M(n,\Bbb R) \cong \Bbb R^{n^2}$, consider unbounded subset of upper-triangular matrices $$A_k = \begin{bmatrix}\lambda_1 & k \\ & \lambda_2 & k \\ & & \ddots & k \\ & & & \lambda_n\end{bmatrix}$$ with $k\to\infty$, $|\lambda_i|\le 2$ as you required, and all other entries $0$.
• As I said, you want to think of matrices as vectors with $n^2$ components, and use the usual vector space structure and norm on $\Bbb R^{n^2}$. So I'm using the Euclidean norm on $\Bbb R^{n^2}$, and the Euclidean length of these matrices $A_k$ is at least $k\sqrt{n-1}$, which goes to infinity as $k\to\infty$. Jan 15, 2014 at 2:08