I want to calculate the strongly convex parameter $\sigma$ for this loss function: $$ l_Z(Z)=||Z-A||^2_F+\lambda tr[Z^TBZ] $$ where $Z\in \mathcal{R}^{n\times m}$, the value of $A,B$ and $\lambda$ are known.
I have tried the Hessian approach, but fail to continue after the gradient: $$ \nabla_Z l=2(\lambda B+I)Z-2A $$
Any help on calculating the Hessian or other approaches would be greatly appreciated.
Since you have computed the gradient already, notice that
$$ \nabla \ell(Z) - \nabla \ell(Z') = 2(\lambda B + I)(Z - Z') $$
Suppose that $\lambda B + I \succeq \sigma I$ for some $\sigma > 0$. Then from the above it is immediate that
$$ \langle \nabla \ell(Z) - \nabla \ell(Z'), Z - Z' \rangle \geq 2\sigma \|Z - Z'\|_F^2 $$
The $2\sigma$ factor above is the modulus of strong convexity.