Given are two matrices:
$\bf A, \bf B$
We know that matrices $\bf A \neq \bf B$ are invertable, symmetric, positive-definite and of full rank. Is it possible to give the formula for following sum of these matrices:
$[\bf A + \lambda\bf B]^{-1} = ?$
where $\lambda$ is a scalar such as $0 < \lambda < 1$.
Best Answer
Assuming that $A+\lambda B$ is also invertible, you can use the Binomial Inverse Theorem:
$[A+\lambda B]^{-1} = A^{-1}-\lambda A^{-1}(I + \lambda BA^{-1})^{-1}BA^{-1}$