X is a symmetric positive definite n by n matrix. This also means it is invertible of course. Consider the matrix Y, which is X, but with an extra row and column at the end: the first n rows of column n+1, and the first n columns of row n+1, are the n diagonals of X. The entry in row n+1, column n+1 of Y is some number T. I am interested in the limit as T approaches infinity of T*(row n+1, columns 1 through n of the inverse of Y). Is there a more utilitarian expression of this? Let's see if I can put this in latex:
$$X=\left( \begin{array}{ccc} x_{11} & x_{12} & .... & x_{1n}\\ x_{12} & x_{22} & .... & x_{2n}\\ .... & .... & .... & .... \\ x_{1n} & x_{2n} & .... & x_{nn}\end{array} \right)$$ $$Y=\left( \begin{array}{ccc} x_{11} & x_{12} & .... & x_{1n} & x_{11}\\ x_{12} & x_{22} & .... & x_{2n} & x_{22}\\ .... & .... & .... & .... & .... \\ x_{1n} & x_{2n} & .... & x_{nn} & x_{nn}\\ x_{11} & x_{22} & .... & x_{nn} & T\end{array} \right)$$ $$Z=Y^{-1}=\left( \begin{array}{ccc} z_{11} & z_{12} & .... & z_{1,n+1}\\ z_{12} & z_{22} & .... & z_{2,n+1}\\ .... & .... & .... & .... \\ z_{1,n+1} & z_{2,n+2} & .... & z_{n+1,n+1}\end{array} \right)$$ And I want to know if there is a useful and simpler expression for:$$\lim_{T \to \infty}(T*[z_{1,n+1}, z_{2,n+1}, .... z_{n,n+1}])$$ PREFERABLY an expression that would not require me to simply approximate taking the limit by using a large value for T; that is what I am looking for, if possible.
Write $Y=\pmatrix{X&\mathbf{u}\\ \mathbf{u}^\top&T}$ and $Z=\pmatrix{\ast&\mathbf{v}\\ \mathbf{v}^\top&z_{n+1,n+1}}$, where the symbol $^\top$ denotes a transpose (not to be confused with the parameter $T$). Using Schur complement, we have $\mathbf{v}=-(X-\mathbf{u}\,T^{-1}\mathbf{u}^\top)^{-1}\mathbf{u}\,T^{-1}$. Therefore $$ \lim_{T\to\infty}T\mathbf{v} =-\lim_{T\to\infty}(X-\mathbf{u}\,T^{-1}\mathbf{u}^\top)^{-1}\mathbf{u} =-X^{-1}\mathbf{u}. $$ In practice, however, it may take a very large $T$ for $T\mathbf{v}$ to approximate $-X^{-1}u$ accurately.