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Suppose that $A \in \mathbb R^{2 \times 2}$ has determinant one, $$ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$

I'm currently working on a problem where I obtained a condition on the quantity $$ \sqrt{\frac{c^2+d^2}{a^2+b^2}}. $$ I was wondering whether this some known quantity which has a specific meaning. Did somebody encounter this quantity before?

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  • $\begingroup$ I think it's the same as $\frac{\sqrt{1 + (ac+bd)^{2}}}{a^{2}+b^{2}},$ but I do not know any way to attach any meaning to the expression, $\endgroup$
    – Geoff Robinson
    Commented Sep 18, 2023 at 19:44
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    $\begingroup$ Trivial observation: it's the ratio between the norms of the two rows. Intuitively it seems related to the condition number: the larger it is, the further $A$ is from a multiple of an orthogonal matrix. $\endgroup$
    – Federico Poloni
    Commented Sep 18, 2023 at 23:20
  • $\begingroup$ No. The matrix with rows equal to $(\cosh\theta ~ \sinh \theta)$ and $(\sinh \theta ~ \cosh\theta)$ can be far from being orthogonal although the ratio of their norm is $1$. $\endgroup$
    – Christophe Leuridan
    Commented Sep 19, 2023 at 11:43

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