The discussion revolves around finding an upper bound \( C \) related to the norm of a matrix \( A \) in the context of vectors \( x \) and \( y \). The original poster seeks to establish a relationship such that \( \| x - Ay \| \leq C \cdot \| x - y \| \).
The discussion is active, with participants sharing their attempts and questioning the assumptions underlying their approaches. Some have provided partial insights into estimating \( C \), while others are looking for clarification or additional hints to advance their understanding.
There is an emphasis on the lack of specific information about the matrix \( A \), which affects the ability to derive a more precise upper bound. Participants are also considering the implications of the entries of \( A \) being constrained to a certain range.
CompuChip said:Some quick-and-dirty trial gives me
C = sup( ||(A - I) v|| )
where I is the identity matrix and the supremum is taken over all vectors v.
I wonder if you can do any better, without more information on A.
CompuChip said:I did
But when all entries of A are between 0 and 1, then you can define ||A|| by
|| A || = max(i, j)( |Aij| )
and use that to get a better estimate.