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I'm starting to look at some lectures on the SVD. The lectures start out by saying that in $$\mathbf{y}=A\mathbf{x}$$ the transformation representing $A$ is only doing rotation and stretching.

  • The matrix for rotation is $$ \begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}$$
  • The matrix for stretching is $$ \begin{bmatrix}\alpha&0\\0&\alpha\end{bmatrix}$$

In the past I've learnt that transformations can be rotations, reflections, scaling, shears, and flattenings. However the idea that any transformation can be decomposed into a rotation and a stretch is new to me. Is there a clear and/or intuitive way of explaining this?

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One major theme in the study of matrix algebra is that of matrix decompositions. The idea is to take a matrix $A$ (either arbitrary or satisfying some reasonable conditions) and decompose it as the product of two or three matrices of a simpler form. For example, the so-called QR decomposition writes an arbitrary matrix $A$ as

$$A = QR$$

Where $Q$ is orthogonal (a composition of rotations + reflections) and $R$ is upper-triangular. In two-dimensions, $Q$ is either rotation or reflection.

In the subject of SVD, an important matrix decomposition is the aptly-named singular value decomposition. This writes an arbitrary matrix $A$ in the form

$$A = UDV^{H}$$

where $D$ is a nonnegative, diagonal matrix whose entries are called the singular values, $U$ and $V$ are orthogonal (unitary in the complex case), and $V^{H}$ is the transpose of $V$ (or conjugate-transpose if complex entries are allowed). A diagonal matrix can be thought of as stretching a basis by the corresponding singular values, and an orthogonal matrix is a rotation or reflection. Thus, this decomposition sees $A$ as a composition of a sort of rotation (transposed), a stretching, and finally another rotation.

You can read more about the singular value decomposition here.

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  • $\begingroup$ Thanks. My question was supposed to be independent of the SVD (but within the context), in that the lecture starts off without mentioning the SVD and claims that every matrix transformation is a combination of rotation and scaling. Reading your answer and looking over the lecture material again, I'm beginning to feel that maybe this is what the SVD shows, that the lecturer's statement was saying what the SVD gives you, rather than giving a premise from which the SVD follows. Is that a fair summary -- that the SVD shows that the decomposition is valid? $\endgroup$ – TooTone Mar 13 '14 at 14:26
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    $\begingroup$ I might be mistaken here (I'm not too knowledgeable about this area), but I think that is correct. SVD is a powerful tool and proving that it exists is tantamount to showing that one's matrix can be decomposed into rotations, scalings, and rotations (in that order). $\endgroup$ – Elchanan Solomon Mar 13 '14 at 19:28

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