1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Norm of matrix
  1. Norm of a matrix? (Replies: 1)

  2. Matrix norm (Replies: 5)

  3. Matrix Norms (Replies: 1)

  4. Norm of matrix (Replies: 1)

  5. Norm of a Matrix (Replies: 4)