The reason the a is scaling the x and the b is scaling the y is because the linear transform matrix is multiplied by every [x; y] point thus:
[LATEX]\begin{bmatrix}
a & 0 \\
0 & b
\end{bmatrix}\begin{bmatrix}
x_i\\
y_i
\end{bmatrix}=\begin{bmatrix}
a \cdot x_i\\
b \cdot y_i
\end{bmatrix}[/LATEX]
The reason the scale appears to be working in the opposite manner can perhaps be seen from this simpler equation (where b = a):
[LATEX]a \cdot x^2 + a \cdot y^2 = 1[/LATEX]
[LATEX]\therefore x^2 + y^2 = \frac{1}{a}[/LATEX]
You can see that a larger scale factor decreases the radius.