I inherited a problem at work and have trouble wrapping my head around it. It doesn't help I'm insufficiently sophisticated with linalg. Help with or pointing to appropriate literature will be much appreciated.
Say, we have $rank-1$ modification of n-rank covariance matrix $C$ and arbitrary $n*1$ vector $v$: $$ C' := C - \frac{vv^T}{<v, C^{-1}v>}$$ Then C' is positive definite with rank n-1.
We can say $C'$ has $n-1$ eigenvectors $e_k (1 \leq k \leq n-1) $ corresponding to $n-1$ positive eigenvalues $\lambda_k $ and matrix $C$ can be decomposed $$C = UU^T $$ where $U$ is matrix with consecutive vector columns: $$ <v,C^{-1}v>^{1/2}u, \sqrt{\lambda_1}e_1, \sqrt{\lambda_2}e_2, ..., \sqrt{\lambda_{n-1}}e_{n-1}$$ Now, here is my first problem - I do not have any sources nor proof nor idea on how can we justify this decomposition of C??
What should follow from above:
Above implies that n-dimensional vector $Z$ given $$ Z = UY$$ where $Y = [Y_1, ... Y_n]$ has covariance matrix $C$. Setting $v = [1,...1]^T$, the standard gaussian random variable $Y_1$ is the variable multiplying the independent and parallel unitary change $(=1)$ in the movements of all marginals $Y_k$
So in the end we could calculate parallel shift on some gaussian variables described via covariance matrix $C$, implied from this covariance matrix, i.e. parallel shift equal to 1 standard deviation would ultimately be:
$$ Y_1(1 sd) = <u, C^{-1}u>^{1/2}u$$
or $99-percentile$ shift
$$ Y_1(99perc) = \phi^{-1}(0.99)<u, C^{-1}u>^{1/2}u$$
where $u = [1,...,1]^T$
Any help or hints for decomposition of C would be much appreciated. Same as pointing to any flaws in this reasoning.