Consider a real, symmetric and positive definite $n\times n$ matrix $\mathbf{K}$, and a $n\times m$ matrix $\mathbf{W}$. $\mathbf{W}$ contains $m$ columns with all zeros except a single entry in each column which is 1. Furthermore $(g\mathbf{B})$ is a $m \times m$ diagonal matrix with positive diagonal elements and $g<0$. Thus, $(g\mathbf{B})$ is negative definite.
With $g \rightarrow -\infty$ I am trying to figure out when the following matrix becomes singular
$$ \mathbf{K}^* = \mathbf{K} + \mathbf{W}(g\mathbf{B})\mathbf{W}^{T} $$
The matrix is large and I don't want to calculate e.g. the lowest eigenvalue of the matrix. Instead I'm looking at when the matrix is invertible, thus by the Woodury formula I get
$$ {\mathbf{K}^*}^{-1} = \mathbf{K}^{-1} - \mathbf{K}^{-1} \mathbf{W} \big[ (g\mathbf{B})^{-1} + \mathbf{W}^T \mathbf{K}^{-1} \mathbf{W} \big]^{-1} \mathbf{W}^T \mathbf{K}^{-1} $$
And I am assuming that the $m \times m$ matrix in the brackets equally must be invertible in order for $\mathbf{K}^*$ to be invertible. However, I seem to be wrong in this assumption as $\mathbf{K}^*$ becomes singular at a different value of $g$ than $\big[ (g\mathbf{B})^{-1} + \mathbf{W}^T \mathbf{K}^{-1} \mathbf{W} \big]$, so:
- Can I relate the definiteness of $\mathbf{K}^*$ to the definiteness of $\big[ (g\mathbf{B})^{-1} + \mathbf{W}^T \mathbf{K}^{-1} \mathbf{W} \big]$ as $g \rightarrow -\infty$?
- Can I say anything about when $\mathbf{K}^*$ becomes singular based on $\big[ (g\mathbf{B})^{-1} + \mathbf{W}^T \mathbf{K}^{-1} \mathbf{W} \big]$?
- Any other suggestions on how to determine the point of singularity of $\mathbf{K}^*$ other than brute force are welcome.