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Prove the Perturbation Estimate

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1. Let A <- C^nxn be invertible and suppose b <- (C^n)_*. Suppose x <- C^n satisfies Ax = b.
Let the perturbations dx, db <- C^n satisfy A dx = db, so that A (x+ dx) = b + db.

(a) Prove the error (or perturbation) estimate

(1 / cond(A)) ( lldbll / llbll ) <= ( lldxll / llxll ) <= cond(A) ( lldbll / llbll ) .

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

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This solution carefully clarifies perturbations.

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