Let { A m } m ≥ 1 be a sequence of n × n complex self-adjoint matrices converging to an n × n matrix A in norm. Then for any s ∈ N show that A m s converges to A s in norm.

Question

Let   {      A    m        }          m      ≥      1       be a sequence of   n  ×  n complex self-adjoint matrices converging to an   n  ×  n matrix A in norm. Then for any   s  ∈      N   show that       A    m    s   converges to       A    s   in norm.

Layne Murillo · Accepted Answer

Convergence in the norm is equivalent to convergence of all the entries of the matrices. So   (      A    m        )          i      j        →      A          i      j       for la i,j. Now   (      A    m          s            )          i      j       is a finite sum of numbers of the type   (      A    m        )          i              k        1              (      A    m        )                  k        1                    k        1              ⋯  (      A    m        )                  k                  s          −          1                    ,      j       so it converges to the (i,j)− element of       A          s

How to show that A_m^s -> A^s if A_m -> A ?

Answered question

Answer & Explanation

New Questions in Precalculus