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.