Suppose we have a parametrized matrix $Z(λ)\in R^{m\times n}$ where $λ\in(a,b)$ and $Ζ(λ)$ is an analytic function of $λ$, e.g. $Z(λ)=λA+(1-λ)B$ where $A,B \in R^{m\times n}$. In general, the (parametrized) SVD of such matrices is given by analytic SVD algorithms, but the value of λ should be computed along with the elements of $U$, $Σ$ and $V^{Τ}$.
Starting from $λΑ+(1-λ)Β=UΣV^{T}$, we do the matrix multiplication to arrive to the system $λa_{ij}+(1-λ)b_{ij}=u_{ik}σ_kv_{jκ}$ where $i=1,...,m$, $j=1,...,n$ and $k=1,...,rank(Ζ(λ))$. Can I compute $λ$ as part of an SVD computation somehow?