= Determinant cone for real symmetric two-by-two matrices
{title2=$\det\begin{pmatrix}z+x&y\\y&z-x\end{pmatrix}=z^2-x^2-y^2$}
The singular real <symmetric matrices> of size two form the double cone $z^2=x^2+y^2$ in these coordinates. Its complement has three <connected components>: $z>\sqrt{x^2+y^2}$ is positive definite, $z<-\sqrt{x^2+y^2}$ is negative definite, and $|z|<\sqrt{x^2+y^2}$ is indefinite. This follows also from the <eigenvalues> $z\pm\sqrt{x^2+y^2}$. The definite cones are convex. In the indefinite region, first move $z$ to zero, then move the nonzero planar vector $(x,y)$ within the punctured plane, proving path-connectedness. Continuous determinant and trace signs prevent joining different regions without crossing singular matrices.
\Image[/past-exam-of-the-metric-spaces-and-complex-analysis-course-of-the-university-of-oxford/2017-symmetric-matrix-cone.png]
{title=The <determinant cone for real symmetric two-by-two matrices> and a radial section}
{description=The cone itself is excluded. The two indefinite wedges visible in the section belong to one connected region in the full three-dimensional space.}
Back to article page
