I need some help to prove the following formula:
$(c \cdot P^{-1} + X \cdot X^T)^{-1}=L\cdot(c \cdot I_n + L^T \cdot X \cdot X^T \cdot L)^{-1}\cdot L^T$
The matrix $X$ has dimension $n \times N$ while the matrix $P$ has dimensions $n \times n$ and is positive (semi?)-definite. The variable $c$ has dimension $1 \times 1$ (non-negative scalar: $c\ge0$). The matrix $L$ is the lower triangular matrix derived from the Cholesky factorization $P=L\cdot L^T$.
Any help to tackle the above simplification is welcome. Thank you
What I tried so far:
$(c \cdot P^{-1} + X \cdot X^T)^{-1}=(c \cdot I_n + P \cdot X \cdot X^T)^{-1}\cdot P=(c \cdot I_n + L \cdot L^T \cdot X \cdot X^T)^{-1} \cdot L \cdot L^T=\ ...$
Equivalently, you want to prove
$$cP^{-1}+XX^T=L^{-T}(cI+L^TXX^TL)L^{-1}$$
Expand the RHS:
$$L^{-T}(cI+L^TXX^TL)L^{-1}=(cL^{-T}+XX^TL)L^{-1}\\=cL^{-T}L^{-1}+XX^T=cP^{-1}+XX^T$$
But note that to take the inverse of $cP^{-1}+XX^T$, it must be invretible. Depends on $XX^T$.