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I'm trying to understand why the subset of $GL(n,\mathbb{R})$ formed by the block-matrices of the following type:

$$\begin{pmatrix} A & B \\ 0 & C \end{pmatrix}$$ where

$$A \in GL(k,\mathbb{R}),~ C \in GL(n-k,\mathbb{R}),~ B \in M(k,n-k,\mathbb{R}).$$

is closed in $GL(n,\mathbb{R})$. I first tried to see this subset as inverse image of a closed subset of $\mathbb{R}$ via the determinant function, and then it would be closed (since $\det$ is continuous), but I couldn't do it. Can you help me?

Alistair Savage
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Br09
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1 Answers1

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First, one needs to be specific about what topology you're interested in. I'm going to assume it is the Zariski topology (although this answer also works for the topology induced by the standard topology on $\mathbb{R}^{n^2}$). The set $\mathrm{GL}(n,\mathbb{R})$ is considered as a subset of $\mathbb{R}^{n^2}$. Then note that your set can be defined as the vanishing set of a set of polynomials. Precisely, it is the vanishing set of the coordinates corresponding to the entries of the lower left block matrix. (Note that if the lower left block is zero and the matrix is invertible, then the blocks on the diagonal must automatically be invertible, so your other conditions are automatic.) Since the vanishing set of a set of polynomials is closed, you're done.

Alistair Savage
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