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Visualising the set of 2×2 positive semidefinite matrices

Recall that a symmetric matrix MRn×n is called positive semidefinite (“PSD”) if, for any xRn, we have xTMx0. 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.