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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.

$$\begin{bmatrix} 3.7 & 0 \\ 0 & 2.1 \end{bmatrix} \times \begin{bmatrix} 0.9510 & -.3090 \\ 0.3090 & 0.9510 \end{bmatrix}$$

If you do the matrix multiplication the opposite way it does do both when the resulting matrix is applied to an ellipse.

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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

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. – Wolphram jonny May 16 '13 at 3:09

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