Visualising the set of 2×2 positive semidefinite matrices

Recall that a symmetric matrix M \in \mathbb{R}^{n \times n} is called positive semidefinite (“PSD”) if, for any x \in \mathbb{R}^n, we have x^{T} M x \geqslant 0. Positive semidefinite matrices occur, for instance, in the study of bilinear forms and as the Gram (or covariance) matrices in probability theory. In the case where n = 2, the space of symmetric matrices is 3-dimensional, and we can actually draw the subset of all positive semidefinite matrices – it looks like the bow of a ship.

It is clear that in the case illustrated below, the PSD matrices form a convex subset. It is easy to show this in general, by observing that the set of all PSD matrices is closed under addition and multiplication by non-negative scalars. The convexity of this set is crucial for the fitting of Mahalanobis distances in metric learning, which is how I got interested PSD matrices in the first place.

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