Prove that N(A*k)⊆N(A*(k+1)) for A∈Rn×n and k∈N.

Question

Prove that N(A^k)⊆N(A^k+1) for A∈R^nxn and k∈N.

Answer 1

A simple proof can be done using only definitions.

First, N(A^k) are the vectors x satisfying A^k x = 0.

Multiplying both sides by A gives A^(k+1) x = 0.

Therefore, if x solves A^k x = 0, it solves A^(k+1) x = 0.

In other words, if x is in the set N(A^k), it is also in the set N(A^(k+1)).