linear algebra question

[gusl]

Given a square matrix B, how does one test whether the sequence defined by B^p converges to the zero matrix, as p->infty?

Comments

[gustavolacerda.livejournal.com]

Proof? Proof by authority would suffice here.

(frozen) (no subject)

[neelk]

Handwavy proof:

Every nonsingular square matrix A can be written as A = P * D * P^(-1), where P are the unit eigenvector matrix, 
and D have the eigenvalues on the diagonal. This is by the eigen decomposition theorem.

Next, we show by induction that A^k = P * D^k * P^(-1)

Case: k = 0 

  A^0 = I 
      = P * P^(-1)
      = P * I * P^(-1)
      = P * D^0 * P^(-1)

Case: k = n+1
  
  A^(n+1) = A^(n) * A 
          = P * D^n * P^-1 * A                by induction hypothesis
          = P * D^n * P^-1 * P * D * P^(-1)   by def of A
          = P * D^n * I * D * P^(-1)
          = P * D^n * D * P^(-1)
          = P * D^(n+1) * P^(-1)

Now, note that if you have a diagonal matrix D, then D^k is also a diagonal matrix whose diagonal elements 
are the k-th powers of the diagonal elements of D. 

So if all of those diagonal elements have absolute values less than 1, then in the limit D^k will go to zero.