Is there any way to solve for $V$ when other matrices in the below equation are known? $V\in\mathbb{R}^{n\times n}$ and is symmetrical, $B\in\mathbb{R}^{n\times (n-1)}$, and $V_{0}\in\mathbb{R}^{(n-1)\times (n-1)}$ is a partition of $V$ without the first row and column. $$V - BV_{0}B^T=X$$
I've solved it numerically using R (where $V$ is eigendecomposed to ensure it is symmertical) but I wonder if there is a quicker way. Here's the function to be minimised for solving this equation,
h<-function(U,X,B){
p<-nrow(X)
U<-matrix(U,nrow=p,ncol=p)
V<-U%*%t(U)
VV<-V-B%*%V[2:(nrow(V)),2:(ncol(V))]%*%t(B)
sum(abs(X-VV))
}
Let $\tilde{{\bf B}} := \begin{bmatrix} {\bf B} & {\bf 0}_n \end{bmatrix}$. We have the following linear matrix equation in symmetric matrix $\bf V$
$$ {\bf V} - \tilde{{\bf B}} {\bf V} \tilde{{\bf B}}^\top = {\bf X} $$
which can be rewritten as a linear system using half-vectorization
$$ \left( {\bf I}_{n^2} - \left( \tilde{{\bf B}} \otimes \tilde{{\bf B}} \right) \right) {\bf D}_n \mbox{vech} ({\bf V}) = {\bf D}_n \mbox{vech} ({\bf X}) $$
where $\otimes$ denotes the Kronecker product and ${\bf D}_n$ is a duplication matrix.
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