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Matrix norms

  1. Dec 18, 2013 #1
    Hi,

    With the following norm inequality:

    ||Av|| ≤ ||A||||v|| implies ||A|| = supv [ ||Av||/||v|| ]

    I understand that sup is the upper bound of a set B, or least upper bound if B is a subset of A, where the upper bounds are elements of both B and A.

    Is this saying that the norm of A is the maximum of the set ||Av||/||v||, where there are multiple vectors v being considered?
     
  2. jcsd
  3. Dec 19, 2013 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    I am puzzled by your use of words here. Of course, you mean "A" and "B" to be sets of numbers. But if a set of numbers has an upper bound, then it has an infinite number of upper bounds so I am puzzled by "sup is the upper bound of a set B, or least upper bound if B is a subset of A". The "sup(B)" is the "least upper bound" in any case.

    And I am puzzled by "where the upper bounds are elements of both B and A." In general upper bounds of sets are NOT in the sets. At most the least upper bound of set A can be in A, in which case it is the maximum of set A. Generally we use the term "sup" specifically to handle those sets that do NOT have a maximum.
     
  4. Dec 19, 2013 #3

    Mark44

    Staff: Mentor

    In your formula above, A is not a set - it's a matrix.
    Yes.
     
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