Consider solving A x = b where A ∈ R n × n and b ∈ R n . Given a perturbation A ^ = A + Δ A, suppose the solution to the system x ^ = x + Δ x. Can I prove the following unequality? ‖ Δ x ‖ / ‖ x ^ ‖ ≤ ‖ A − 1 ‖ ‖ Δ A ‖

Question

Consider solving   A  x  =  b where   A  ∈        R          n      ×      n       and   b  ∈        R    n  . Given a perturbation       A    ^    =  A  +  Δ  A, suppose the solution to the system       x    ^    =  x  +  Δ  x. Can I prove the following unequality?  ‖      Δ    x    ‖  /  ‖      x    ^    ‖  ≤  ‖      A          −      1        ‖  ‖  Δ  A  ‖

persstemc1 · Accepted Answer

use       A    ~   and       x    ~   for pertubations.You can start by using   A  x  =  b and   (  A  +      A    ~    )  (  x  +      x    ~    )  =  b Expanding the second equation gives you   A  x  +      A    ~    x  +  A      x    ~    +      A    ~        x    ~    =  b. You use your knowledge of   A  x  =  b and arrive at  A      x    ~    =  −      A    ~    x  −      A    ~        x    ~    .Since   A is invertible (otherwise, there would be no unique solution   x to   A  x  =  b), you can determine       x    ~    =  −      A          −      1            A    ~    x  −      A          −      1            A    ~        x    ~  . By taking the norm of both sides, you arrive at  ‖      x    ~    ‖  =  ‖      A          −      1            A    ~    (  x  +      x    ~    )  ‖  .Taking the sub-multiplicativity into account, you get exactly where you want to be!

Consider solving A x = b where A ∈ <mi mathvariant

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