Let A be a n × n square matrix, and ‖ A ‖ 1 := max j = 1 n ∑ i = 1 n | A i , j | and ‖ A ‖ max := max i , j | A i , j | 1)Does ‖ A ‖ 1 ≤ 1 imply that ‖ A ‖ max ≤ 1?2)Is there any relation between ‖ A ‖ 1 and ‖ A ‖ max ?

Question

Let A be a   n  ×  n square matrix, and  ‖  A      ‖    1    :=      max          j      =      1              n            ∑          i      =      1              n            |        A          i      ,      j            |  and  ‖  A      ‖          max        :=      max          i      ,      j            |        A          i      ,      j            |  1)Does   ‖  A      ‖    1    ≤  1 imply that   ‖  A      ‖          max        ≤  1?2)Is there any relation between   ‖  A      ‖    1   and   ‖  A      ‖          max      ?

Derick Ortiz · Accepted Answer

Since      |        A          i      ,      j            |    ≤      max          1      ≤      i      ≤      n            |        A          i      ,      j            |    ,for each   1  ≤  j  ≤  n, then      ∑          i      =      1              n            |        A          i      ,      j            |    ≤      ∑          i      =      1              n            max          1      ≤      i      ≤      n            |        A          i      ,      j            |    =  n      max          1      ≤      i      ≤      n            |        A          i      ,      j            |  Therefore      max          1      ≤      j      ≤      n            ∑          i      =      1              n            |        A          i      ,      j            |    ≤  n      max          1      ≤      i      ,      j      ≤      n            |        A          i      ,      j            |    .That is  ‖  A      ‖          1        ≤  n  ‖  A      ‖          ∞        .    (  1  )The estimate (1) is sharp. Meaning the factor n cannot be replaced by a smaller positive constant. To see this consider the matrix A with       A          i      ,      j        =  1 for which (1) is actually an equality.Now, does   ‖  A      ‖          1       control   ‖  A      ‖          ∞      ?We have that      max          1      ≤      i      ≤      n            |        A          i      ,      j            |    ≤      ∑          i      =      1              n            |        A          i      ,      j            |    ⟹      max          1      ≤      j      ≤      n            max          1      ≤      i      ≤      n            |        A          i      ,      j            |    ≤      max          1      ≤      j      ≤      n            ∑          i      =      1              n            |        A          i      ,      j            |  So, it is true that  ‖  A      ‖          ∞        ≤  ‖  A      ‖          1          (  2  )Again the constant one in (2) is sharp. When   A  =      I          n      , the unit matrix, we have an equality in (2).

Does norm(A)_1 <= 1 imply that norm(A)_(max) <= 1?

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