0

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.

kampfkoloss
  • 21
  • Your observation that a vector is a one-column matrix is correct but - in my opinion - misses the point. In $Ax=b$ the question is: How does $x$ change due to errors? But - it seems - in matrix multiplication $C=AB$ the matrix $B$ is known. – g g Sep 28 '22 at 16:48
  • Thanks @gg . In the matrix multiplication $A$ and $B$ are known, but $C$ is "unknown". So the question would be "How does $C$ change due to numerical inaccuracies in $A$"? That is why I was wondering if condition number of $A$ is relevant for matrix multiplication as well. However, this question is perhaps not relevant at all since matrix multiplication works on one row at the time of $A$? – kampfkoloss Sep 29 '22 at 08:57

0 Answers0