The approximation A-1ε ≈ A-1 - εA-1BA-1 for a perturbed matrix Aε = A + ε B is derived using a series expansion while neglecting the higher-order terms of ε.
The student is asking how to approximate A-1ε given A-1 for a perturbed matrix Aε = A + ε B, where ε is a small value. This can be derived using the definition of the inverse of a sum of matrices and perturbation theory. We assume that A is invertible and ε is sufficiently small such that Aε remains invertible.
To find Aε-1, we use the formula (A + ε B)-1 and apply a series expansion, neglecting higher-order terms of ε since it is small. To the first order, this gives us the approximation A-1ε ≈ A-1 - ε A-1B A-1, illustrating that the perturbed inverse can be approximated by subtracting ε times the product of A-1, B, and A-1 from A-1.
Conclusion: With the help of elementary matrix operations and perturbation theory, a useful approximation for the inverse of the perturbed matrix is obtained, which is particularly valuable when dealing with small perturbations in practical applications.