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Norm of matrix

  1. Mar 2, 2008 #1
    1. The problem statement, all variables and given/known data
    Let A = [a_{ij}] be a mxn matrix. Show that max[tex]_{ij}[/tex]|a[tex]_{ij}[/tex]| ≤ ‖A‖ ≤ √(∑[tex]_{ij}[/tex]|a[tex]_{ij}[/tex])|

    2. Relevant equations

    3. The attempt at a solution

    By the definition ‖A‖=max_{||x||≤1}‖A(x)‖ for all x ∈ Rⁿ.So, ‖A‖≥‖A∘(x₁,..,x_{n})[tex]^{T}[/tex]‖ for x = (0,...,1,...0) with 1 is in the i[tex]^{ij}[/tex] position and so ‖A‖ ≥ ‖A∘(x₁,..,x_{n})[tex]^{T}[/tex]‖ = ||(a[tex]_{i1}[/tex],a[tex]_{i2}[/tex],...,a[tex]_{ij}[/tex])|| = √(a[tex]_{i1}[/tex][tex]^{2}[/tex]+...+a[tex]_{in}[/tex]) ≥ max[tex]_{ij}[/tex]|a[tex]_{ij}[/tex]|.
    I do not know what how to do the upper bound.
    Last edited: Mar 2, 2008
  2. jcsd
  3. Mar 2, 2008 #2
  4. Mar 2, 2008 #3


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    Science Advisor
    Homework Helper

    That's sort of hard to read - do you want to prove that [itex]\| A \|^2 \leq \sum_{ij} |a_{ij}|^2[/itex]?

    If so, the Cauchy-Schwarz inequality will be very useful.
  5. Mar 2, 2008 #4
    Thank you for replying!
    I will think about this.
    Last edited: Mar 2, 2008
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