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!
Best Answer
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
$$(A+mI)^{-1}=P\operatorname{diag}(\frac{1}{a_i+m})P^{-1}$$