Plotting 2x2 SPD matrices

Introduction

In this short note, I want to show you how we can plot data points defined in the symmetric positive definite (SPD) manifold. This can be useful when doing illustrative examples or pedagogical explanations for algorithms applied to this sort of data. I will focus on the case of $2 \times 2$ matrices because it is the only situation in which the plots are exact, i.e. without any distortion. For higher dimensional data, one should recur to one of the several algorithms for non-linear dimensionality reduction (e.g. t-SNE, MDS, PCA, Laplacian eigenmaps, etc). I was inspired to write this note after trying to reproduce Figure 1 from this article for my presentation about pyriemann for the BCI meeting 2023 [link].

SPD matrices

The most natural way of parametrizing a $2 \times 2$ SPD matrix is to write

which clearly can be displayed in a three-dimensional plot. Sweeping different values $(a_i, b_i, c_i)$ will yield different matrices matrices $\boldsymbol{X}_i$, but to ensure that they are all SPD, they have to respect that $a_i c_i - b_i^2 > 0$.

Respecting such condition is not necessarily very intuitive, so a different parametrization for the SPD matrices can be

where $\theta \in [0, 2\pi]$ and $\lambda_1, \lambda_2 > 0$ will ensure that $\boldsymbol{X}$ is SPD.

What to plot?

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An example

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import numpy as np
from scipy.stats import multivariate_normal
import matplotlib.pyplot as plt

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Conclusion

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