How to prove the induced 1-norm satisfies a specific equation?

[peripatein]

Hi,

Homework Statement

I'd like to show that the induced 1-norm satisfies: ∥A∥₁=max_1≤j≤n∑[i=1 to n] |a_i,j|

Homework Equations

The Attempt at a Solution

I realize the sum ∑[i=1 to n] |a_i,j| is basically ||Ax|| where x is the j-th basis. I also know that ||A||₁=max_||x||1=1||Ax||. And since ||x||₁=1, ∑[i=1 to n]|x_i|=1.
But I am not sure how to assemble all that together. Would anyone please advise?

 

[kurt.physics]

The answer is as follows: By definition, the 1-norm of a matrix A is given by ||A||1=max||x||1=1||Ax||. Let x be the j-th basis vector, such that ||x||1=1 and x=(0,0,...,1,0,...) where the 1 is at the j-th place. Then ||Ax||=∑|ai,j|=∑[i=1 to n]|ai,j|. Since ||x||1=1, we can conclude that ∥A∥1=max1≤j≤n∑[i=1 to n] |ai,j|.