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I don't understand why if you multiply a scaling matrix with rotation matrix that the resulting matrix, when applied to a shape like an ellipse, only gets scaled and does not get rotated.

$$​​\left( \matrix{3.7 \ \ 0\\ 0 \ \ 2.1}\right)\times\left( \matrix{0.9510 \ \ -0.3090 \\ 0.3090 \ \ 0.9510}\right)$$

If you do the matrix multiplication the opposite way it does do both when the resulting matrix is applied to an ellipse. Sorry about the matrices I don't know how to format them in Latex.

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Could you give an example? –  Memming May 16 '13 at 2:44
    
This is obviously false - take the unit scale matrix for example. –  nbubis May 16 '13 at 2:49
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The only way you could be "right" is for a rotation with $\theta=n\pi/2$ ($n \in \mathbb{Z})$. If $n$ is odd you need a scaling that "reverses" the shape of the ellipse, in such a way that the major axis becomes the minor axis, and the minor axis becomes the major axis, with the same proportions as the original ellipse.

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you just added an example, and say that the reverse is an actual rotation and scaling. That is surely wrong, perhaps some mistake in your calculations. Although, a look at your matrix looks that you are scaling the two axes by a different factor. That is not called scaling as far as I know. –  julian fernandez May 16 '13 at 3:09
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