Cfrgtkky

Suppose you have a 2x2 real-valued matrix, $mathbf{M}$. If you perform a singular value decomposition (SVD), then Wikipedia and the internet tell me that this can be understood geometrically as a decomposition of $mathbf{M}$ into a rotation, scaling and second rotation of the form:

$$mathbf{M} = mathbf{U S V}^mathrm{T}$$

where the $mathrm{T}$ denotes transpose. Here, $mathbf{V}^mathrm{T}$ is the first rotation, $mathbf{S}$ is the scaling matrix and $mathbf{U}$ is the last rotation.

Wikipedia has this nice little graphic:

So, what I am trying to do is essentially reproduce this graphic in MATLAB. I have a unit circle represented as:

$$mathbf{C} = begin{bmatrix}
1&0 \ 0
& 1
end{bmatrix}$$

This can be plotted in MATLAB like this:

I then have some arbitrary shearing matrix which is given by:

$$mathbf{M} = begin{bmatrix}
0.5&4 \ 1
& 1.5
end{bmatrix}$$

which represents an ellipse which can also be plotted in MATLAB:

So far so good because I can reproduce the top two images in the Wikipedia article.

If I do the SVD for $mathbf{M}$ I can get:

$$mathbf{U} = begin{bmatrix}
-0.9239&-0.3827 \ -0.3827
& 0.9239
end{bmatrix}$$

$$mathbf{S} = begin{bmatrix}
4.3523&0 \ 0
& 0.7467
end{bmatrix}$$

$$mathbf{V} = begin{bmatrix}
-0.1941&-0.9810 \ -0.981
& -0.1941
end{bmatrix}$$

Now, can someone explain what I need to do to produce the bottom two images from the Wikipedia article?

I thought that I would just be able to plot the columns of $mathbf{V}^mathrm{T}mathbf{C}$ to get the rotated unit vectors for the bottom left image like so:

Next, I would plot the ellipse of $mathbf{SC}$ to get the bottom right stretched image and I can apply $mathbf{SV}^mathrm{T}$ to the unit vectors to get:

This is clearly wrong though because the major axis of the dashed ellipse is much longer than the major axis of the solid ellipse representing $mathbf{M}$. If I rotate the dashed ellipse, it will not match the solid ellipse.

What am I doing wrong?