Main idea

Let E=(I+Ak)(I+A1)R be the error matrix.

Observe

ΦAi=(I+Ai+1)(I+Ak)E(I+A1)(I+Ai1)

Since Ajτ<1, each term (I+Aj) is invertible!

Hence, gradient can only vanish if E=0.

Let E=(I+Ak)⋯(I+A1)−R be the error matrix.