Proving the perturbation estimate

Question

Question:

Prove the Perturbation Estimate

1. Let A∈ Cnxn be invertible and suppose b∈ Cⁿ_*. Suppose x∈Cn satisfies Ax = b.
Let the perturbations δx, δb∈Cn satisfy Aδx = δb, so that A (x+ δx)+ δb.

(a) Prove the error (or perturbation) estimate

1/cond(A) ( llδbll / llbll )≤( llδxll / llxll )≤cond(A) (llδbll/llδll ) .

(b) Show that for any invertible matrix A, the upper bound for ( llδxll / llxll ) above can be attained for suitable choices of b and δb