Discussion Overview
The discussion revolves around the relationship between matrix norms induced by different vector norms, specifically questioning whether the equality of the induced matrix norms follows from the equality of their action on vectors.
Discussion Character
- Technical explanation, Debate/contested
Main Points Raised
- One participant questions if ||A||_{a} = ||B||_{b} can be concluded from the condition ||Ax||_{a} = ||Bx||_{b} for any x in \mathbb{R}^n.
- Another participant asserts that the definition of the matrix norm can be used to demonstrate the equality, although this claim lacks further elaboration.
- A third participant provides the definition of the matrix norm induced by a vector norm, noting a concern about the differing values of ||x||_{a} and ||x||_{b} in the supremum calculation.
- A later reply seeks clarification on whether the discussion pertains to operator norms.
Areas of Agreement / Disagreement
Participants have not reached a consensus; there are differing views on whether the equality of matrix norms can be established from the given condition, and concerns remain regarding the implications of differing vector norms.
Contextual Notes
The discussion highlights potential limitations in the approach, particularly regarding the treatment of the denominator in the supremum and the implications of using different vector norms.