Singular-value decomposition can be used to decompose any affine transformation into a sequence involving rescaling along some properly oriented set of axes, followed by a rigid-body transformation of the rescaled result. Conceptually, this means that the object being transformed can be viewed as residing within a special reference frame. Inside this special reference frame, the object is simply rescaled. Meanwhile, the reference frame itself is subjected to a rigid-body transformation. Viewed in this way, it does not matter whether the reference frame is rotated before the object within it is rescaled or after the object within it is rescaled. Indeed, the two operations can even be viewed as occurring simultaneously.
For the numerical example, singular-value decomposition gives
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