Maths - Types of 2 Dimensional Matrix Transforms

Here are some examples of 2 dimensional transforms and their corresponding 2x2 matrices.

I have noted whether the matrix is symmetric or anti-symmetric across the leading diagonal. As you can see, rotation is anti-symmetric across the leading diagonal but symmetric along the diagonal, reflection is symmetric across the leading diagonal but anti-symmetric along the diagonal. This is not necessarily the case for 3D transforms here.

1 0
0 1

Identity

(symmetric)

a 0
0 a

Scale

(symmetric)

0 1
1 0

Swap x and y axes, which is the same as reflecting in a 45° line

(symmetric)

0 -1
1 0

Swap x and y axes and invert y, which is the same as rotating by 90°

(anti-symmetric)

cos(θ) -sin(θ)
sin(θ) cos(θ)

Rotate by angle θ

(anti-symmetric - but symmetric along the leading diagonal)

-1 0
0 1
Reflection in x axis
-x2 + y2 - 2 * x * y
- 2 * y * x -y2 + x2

Reflection in line x,y (see this page)

(symmetric - but anti-symmetric along the leading diagonal)

   

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