 Projection onto a positive semidefinite trace ball

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 ${\sum }_i({\mu }_i-\rho {)}_{+}=s$.