Is there any special properties about the inverse of a unit lower triangular matrix?
I'm trying to prove this: $$L^{-1}=I_n + N + N^2 + ... + N^{n-1}$$
where $L$ is a unit lower triangular matrix and $L = I_n -N$
So obviously, $N$ must also be a lower triangular matrix with all zeroes on the diagonal, but I don't know how this helps.
I only getting as far as showing that $L + N = (L + N)^2 = ... = (L + N)^{n-1}$
Therefore, $$(n-1)I_n = N + N^2 + N^3 + ... + N^{n-1} + L + L^2 + L^3 + ... + L^{n-1} + G(N, L)$$ which is equivalent to $$nI_n - (L + L^2 + L^3 + ... + L^{n-1} + G(N, L)) = I_n +N +N^2 + N^3 + ... + N^{n-1}$$ where $G(N, L)$ is still some function of N and L)
I'm stuck here and not sure how to show that the LHS is actually $L^{-1}$
Anyone has any suggestion? Or I'm completely off track using this method?
Check that $N^n=0$ where $n\times n$ is the size of the matrix. A direct calculation then shows that $(I_n-N)(I_n+N+N^2+\cdots N^{n-1})= (I_n+N+N^2+\cdots N^{n-1})(I_n-N)=I_n$ (use distributivity).