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I have the following linear map $\mathbb{R}^2\to\mathbb{R}^2:$

$$\begin{pmatrix}x\\y\end{pmatrix}\mapsto \begin{pmatrix}9y-5x\\7y-4x\end{pmatrix}$$

I am asked to describe this geometrically.

I can see what is happening: the upper left quadrant is being squeezed into the upper right quadrant, in between the lines $y=\frac{7}{9}x$ and $y=\frac{4}{5}x$. Likewise the lower right quadrant is scrambled in between the same lines in the lower left quadrant. The other quadrants are stretched in the obvious way.

I can't find the proper terms to describe this well though. A rotation followed by a squeeze? Thanks to anyone who responds.

(I can upload an image if anyone is interested, but the lines are so close that it will not be very clear)

user118224
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3 Answers3

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Suppose you have the map $$ T_2 \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 9y + x \\ y \end{pmatrix}. $$ Then this is called a horizontal shear. There is more information here: Shear mapping. Vertical shear is similarly defined. Now consider three additional maps $$ T_1 \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -5x \\ y \end{pmatrix} \\ T_3 \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ -\frac{1}{5}y \end{pmatrix} \\ T_4 \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y + \frac{4}{5}x \end{pmatrix}. $$ Then the given map is just $$ T_4T_3T_2T_1 \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 9y - 5x \\ 7y - 4x \end{pmatrix}. $$ So the map is a combination of horizontal scaling followed by horizontal shear followed by vertical scaling and finally a vertical shear. Note that you can write transformations like this in terms of simpler transformations in many different ways. So the one I gave is just one possible way.

Pratyush Sarkar
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Since you have a matrix transformation here; $$\begin{bmatrix}x\\y\\\end{bmatrix}\mapsto\begin{bmatrix}-5&9\\-4&7\\\end{bmatrix}\begin{bmatrix}x\\y\\\end{bmatrix},$$ you can split up A as a sum of transformation matrices, where $A=\begin{bmatrix}-5&9\\-4&7\\\end{bmatrix}$ $$A=\begin{bmatrix}-5&9\\-4&7\\\end{bmatrix}=\begin{bmatrix}1&9\\0&1\\\end{bmatrix}+\begin{bmatrix}-7&0\\0&5\\\end{bmatrix}+\begin{bmatrix}1&0\\-4&1\\\end{bmatrix}$$ Recall that shear transformation matrices look like this: $$\begin{bmatrix}1&k\\0&1\\\end{bmatrix}(\text{x-axis})\text{ or }\begin{bmatrix}1&0\\k&1\\\end{bmatrix}\text{(y-axis)}$$ And scaling matrices look like this: $$\begin{bmatrix}s_x&0\\0&s_y\\\end{bmatrix}$$ So then, $\begin{bmatrix}1&9\\0&1\\\end{bmatrix}$ and $\begin{bmatrix}1&0\\-4&1\\\end{bmatrix}$ are shear transformations on the x and y axes respectively and $\begin{bmatrix}-7&0\\0&5\\\end{bmatrix}$ is a scaling transformation.

Not an easy time to see geometrically as simply $A$, but hopefully you can see the transformations better one step at a time.

Eleven-Eleven
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A QR decomposition for $A=\begin{bmatrix}-5&9\\-4&7\end{bmatrix}$ is

$$\begin{bmatrix}\frac{-5}{\sqrt{41}}&\frac{-4}{\sqrt{41}}\\\frac{4}{\sqrt{41}}&\frac{-5}{\sqrt{41}}\end{bmatrix}\begin{bmatrix}\sqrt{41}&\frac{-73}{\sqrt{41}}&\\0&\frac{1}{\sqrt{41}}&\end{bmatrix}$$

and we can additionally split out a dilation like so:

$$\begin{bmatrix}\frac{-5}{\sqrt{41}}&\frac{-4}{\sqrt{41}}\\\frac{4}{\sqrt{41}}&\frac{-5}{\sqrt{41}}\end{bmatrix}\begin{bmatrix}\sqrt{41}&0\\0&\sqrt{41}\end{bmatrix}\begin{bmatrix}1&\frac{-73}{41}&\\0&\frac{1}{41}\end{bmatrix}$$

In the last decomposition, let's call the matrices $R,D,T$ respectively.

This decomposition shows that $A$ is the triangular mapping $T$ followed by the dilation $D$ and then the rotation $R$.

The triangular mapping keeps the $x$ axis fixed and swings the $y$-axis around to a new direction. The dilation and reflection are self-explanatory, I hope.

You can also note that the rotation is a proper rotation with no reflection, since its determinant is positive. The triangular mapping swings the $y$ axis out into the third quadrant. The dilation in this case expands the plane by a factor of about 6.7. The rotation is by about 141 degrees counterclockwise.

Analyzing it with singular value decomposition instead, we find two singular values of about $13.08$ and $0.08$. This says that although one direction is contracting, the orthogonal direction is expanding.

rschwieb
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