(3) Suppose an n X n matrix A has n independent eigenvectors v1, . , on, with associated eigenvalues A1, . , An. (a) Show that if all the eigenvalues…

  1. Suppose an n × n matrix A has n independent eigenvectors v1, . . . , vn, with associated eigenvalues λ1, . . . , λn.
  2. (a) Show that if all the eigenvalues have absolute value less then 1, then for everyx∈Rn wehaveAkx→0ask→∞.
  3. (b) Show that if one of the eigenvalues (say, λ1) has absolute value greater than 1, thenthenthereexistsx∈Rn suchthat|Akx|→∞ask→∞.

(3) Suppose an n X n matrix A has n independent eigenvectors v1, . . . , on, with associatedeigenvalues A1, . . . , An. (a) Show that if all the eigenvalues have absolute value less then 1, then for every$61K” wehaveAkzc—>0ask—>oo. (b) Show that if one of the eigenvalues (say, A1) has absolute value greater than 1,then then there exists a: E R" such that |Akzl —> 00 as k —> oo.