Suppose that $A$ is a symmetric positive definite matrix and assume its dimension $n$ is large. Let $I$ be the $n \times n$ identity matrix and $m \neq 0$ be a scalar. I'm interested in computing as many of the following as possible:
- det$(A + mI)$
- $(A + mI)^{-1}$
- $(A + mI)^{-1}B$, where $B$ is an $n \times m$ matrix
- the Cholesky decomposition of $A + mI$,
I'd like to do this for many values of $m$. However, because $n$ is large, I'd like to know if there is some update trick based on det$(A)$, $A^{-1}$, and the Cholesky decomposition of $A$. $A$ will likely not be sparse. I've been researching this for quite a while and the results haven't been all that encouraging.
Any help, hints, suggestions, or references would be much appreciated!
Use the fact that symetric positive definite matrix is similar to a diagonal matrix with positive elements. More specifically, $$ P^{-1}AP = \operatorname{diag}\{a_1,...a_n\} $$ where $a_i>0$ for $i \in [1, n]$, diag{$a_1,...a_n$} is diagonal matrix, and $P $ is invertible. So We have
\begin{align} \det(A+mI)&=\det(P^{-1}(A+mI)P) \\ &=\det(P^{-1}AP+P^{-1}(mI)P) \\ &=\det(P^{-1}AP+mI) \\ &=\prod \limits_{i=1}^{n}(a_i+m) \end{align}
Since $$ P^{-1}(A+mI)P = \operatorname{diag}(a_i+m) $$ And $$ \operatorname{diag}(a_i+m)^{-1}=\operatorname{diag}(\frac{1}{a_i+m}) $$ where $a_i+m\neq0$
Thus \begin{align} (P^{-1}(A+mI)P)^{-1}&=P(A+mI)^{-1}P^{-1} \\ &=\operatorname{diag}(a_i+m)^{-1} \\ &=\operatorname{diag}(\frac{1}{a_i+m}) \end{align}
And
$$(A+mI)^{-1}=P\operatorname{diag}(\frac{1}{a_i+m})P^{-1}$$