Let both $A,B\in\mathbb{R}^{n \times n}$ and positive definite. I know the condition number for $A$ in $Ax=b$ quantifies the sensitivity in this problem. What I am wondering about is if the condition number is important in the matrix multiplication $C=AB$? That is, can you measure how sensitive $C$ is to small changes in $B$ by the condition number of $A$?
My (limited) understanding is that the condition number of the matrices should be relevant for matrix multiplications since $C[:,i]=AB[:,i]$. Here, $B[:,i]$ means the $i$-th column vector of $B$. Is this understanding correct?
This old thread Condition number matrix matrix multiplication was not so clear for me in this regards.