2D Rotation matrices (2): properties

Antonio Sala, UPV

Difficulty: ** ,       Relevance: ,      Duration: 18:22

Summary:

In this video, we examine the properties of the 2D real rotation matrices which we derived in video [rot2d1EN] from a sum-of-angles argumentation. We will discuss:

1. Orthogonality: $R^{T}R=I$ is verified. That entails that inverting and transposing is the same, and columns (or rows) of 2D rotation matrices form an orthonormal basis. It also entails ‘metric and angle preservation’ (intuitively: rotations change neither lengths nor shape of things).

2. Unit determinant ($+1$): orientation preservation, rotations are not reflections (whose determinant is $-1$).

3. Eigenvalues and eigenvectors: they are complex-valued; so, maybe extending the analysis to complex-valued vectors will give further insight. It does: quaternions will be discovered there.

4. Commutativity: 2D rotations fulfill $R({𝜃}_1)R({𝜃}_2)=R({𝜃}_2)R({𝜃}_1)=R({𝜃}_1+{𝜃}_2)$. This does not hold in 3 or more dimensions.

5. Products of rotation matrices mirror addition of angles. So, this suggests a logarithm-like meaning for angles, or an exponential-like meaning for rotations. Indeed, this is the basic idea behind the exponential map to be discussed in a forthcoming video.

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.