Positive semidefinite cone

The positive semidefinite matrices form a convex cone because nonnegative combinations preserve nonnegative quadratic forms.

The set of real positive semidefinite matrices with unit diagonal. Equivalently, its members are Gram matrices of unit vectors, by the real spectral theorem. Every two-by-two principal minor gives $|X_{ij}|\le 1$, so the set is bounded; it is also closed, hence compact in finite dimension. This gives attainment of linear objectives used in semidefinite programming.

For an integer $0\le k\le n$, the fantope is the convex hull of rank-$k$ orthogonal projection matrices in ${\mathbb{R}}^n$. Equivalently, it consists of real symmetric matrices with eigenvalues in $[0,1]$ and matrix trace $k$. To prove the equivalence, apply the spectral theorem for real symmetric matrices and express the eigenvalue vector as a convex combination of the zero-one extreme points of the capped simplex. The sum of the largest eigenvalues is its support function.

For a positive semidefinite matrix, a trace bound is a bound on the sum of its nonnegative eigenvalues.

The positive semidefinite trace ball of radius $s$ isIt is closed and convex, and $\Vert Z{\Vert }_F\le \mathrm{tr}Z\le s$ for every $Z\in {\mathcal{S}}_s$.

If $M⪰0$ has eigenpairs $({\mu }_i,v_i)$ and $\mathrm{tr}M>s$, then its Frobenius projection onto ${\mathcal{S}}_s$ iswhere $\rho >0$ is uniquely determined by ${\sum }_i({\mu }_i-\rho {)}_{+}=s$.