Proving Norm of Matrix Inequality for Homework

[Gtay]

Homework Statement

Let A = [a_{ij}] be a mxn matrix. Show that max_{ij}|a_{ij}| ≤ ‖A‖ ≤ √(∑_{ij}|a_{ij})|

Homework Equations

The Attempt at a Solution

By the definition ‖A‖=max_{||x||≤1}‖A(x)‖ for all x ∈ Rⁿ.So, ‖A‖≥‖A∘(x₁,..,x_{n})^{T}‖ for x = (0,...,1,...0) with 1 is in the i^{ij} position and so ‖A‖ ≥ ‖A∘(x₁,..,x_{n})^{T}‖ = ||(a_{i1},a_{i2},...,a_{ij})|| = √(a_{i1}^{2}+...+a_{in}) ≥ max_{ij}|a_{ij}|.
I do not know what how to do the upper bound.

 

[Gtay]

Anyone?

 

[morphism]

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

If so, the Cauchy-Schwarz inequality will be very useful.

 

[Gtay]

Thank you for replying!
I will think about this.