Show $y_i (C) = y_i (A)$ for two matrices $A$ and $C$

Question

Let $A$ be a symmetric positive definite matrix. Suppose that $A$ can be written as $A = C C^T$, where $C$ is a lower triangular matrix. Define the function $y_i (A) = min \{j | a_{ij} \neq 0 \}$, for $i = 1, ..., n$.

Show that $y_i (C) = y_i (A)$, for $i = 1, ..., n$.

Since $A$ is symmetric, we have $y_i (A^T) = y_i ((C C^T)^T) = y_i (C C^T) = y_i (A)$.

This is what I have now. I don't see how I can get $y_i (C) = y_i (A)$. Some help would be great.

Answer 1

Translating into English, the proposition $y_i(C)=y_i(A)$ means that:

$(\ast)$ on each row, the first nonzero entries of $C$ and $A$ occur at the same position.

Since $A$ is positive definite, $C$ must be nonsingular and hence $c_{ii}\ne0$ for each $i$. Therefore $c_{ii}$ and $a_{ii}$ are always nonzero. So, in order to prove $(\ast)$, it suffices to prove that

(#) When $j<i$, $a_{i1}=a_{i2}=\cdots=a_{ij}=0$ if and only if $c_{i1}=c_{i2}=\cdots=c_{ij}=0$.

Now, recall that $c_{ii}\ne0$ and $a_{ij}=\sum_{k=1}^j c_{ik}c_{jk}$ when $j<i$. You may go from here.