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The group $\mathrm{GL}_n(\mathbb{R})$ inherits a metric under the injection $$\mathrm{GL}_n(\mathbb{R}) \hookrightarrow \mathbb{R}^{n^2}$$.

Is there a natural/geometric way of viewing this metric on $\mathrm{GL}_n(\mathbb{R})$? What does is mean for two linear transformations to be close? What properties do two "close" linear transformations share?

Rdrr
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    The general linear group is not a linear space. It does not contain zero, for example. However it is a subspace of $n^2$-dimensional Euclidean space, so it does inherit a metric. Let's just not call it a norm. – ziggurism Jan 15 '18 at 23:24
  • The natural geometry on $GL_n(\mathbf R)$ does not come from the embedding into the Euclidean space, but rather, from the Lie group structure. – tomasz Jan 16 '18 at 01:55

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Viewing it as a norm on $n^2$-dimensional Euclidean space (rather than restricting to the general linear group), it is called the Frobenius norm (or inner product). For matrices with good spectral properties (and hence I think for all matrices), it coincides with the Schatten norm, which is the diagonal length of image of the unit box under multiplication by the matrix. Compare that with the operator norm, which is the length of the longest side of the image of the unit box.

See the excellent answer by Eric Naslund here.

ziggurism
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