What is the Maximum Norm Proof for Matrix A?

[Amer]

Prove that for

$A \in \mathbb{R}^{n\times n} $
$$||A||_{\infty} = \text{max}_{i=1,...,n} \sum_{j=1}^n |a_{ij} |$$

I know that
$||A||_{\infty} = \text{max} \dfrac{||Ax||_{\infty} }{||x||_{\infty}} $

such that $x \in \mathbb{R}^n$

any hints

 

[Dustinsfl]

Prove that for

$A \in \mathbb{R}^{n\times n} $
$$||A||_{\infty} = \text{max}_{i=1,...,n} \sum_{j=1}^n |a_{ij} |$$

I know that
$||A||_{\infty} = \text{max} \dfrac{||Ax||_{\infty} }{||x||_{\infty}} $

such that $x \in \mathbb{R}^n$

any hints

I believe you can start with $||A||_p$ the p norm and take the limit as $p\to\infty$ to prove the problem.
Isn't the infinity norm just the max of $|a_i|$ not the sum of them?

 

[Amer]

I believe you can start with $||A||_p$ the p norm and take the limit as $p\to\infty$ to prove the problem.
Isn't the infinity norm just the max of $|a_i|$ not the sum of them?

for a vector it is the max of $|a_i|$

 

[Opalg]

Prove that for

$A \in \mathbb{R}^{n\times n} $
$$||A||_{\infty} = \text{max}_{i=1,...,n} \sum_{j=1}^n |a_{ij} |$$

I know that
$||A||_{\infty} = \text{max} \dfrac{||Ax||_{\infty} }{||x||_{\infty}} $

such that $x \in \mathbb{R}^n$

any hints

You might find it easier to use the equivalent definition $\|A\|_\infty = \max \{\|Ax\|_\infty : \|x\|_\infty \leqslant 1\}.$ For $\|x\|_\infty \leqslant 1$, show that $$\|Ax\|_\infty = \max_{1\leqslant i\leqslant n}|(Ax)_i| = \max_{1\leqslant i\leqslant n}\Bigl| \sum_{j=1}^n a_{ij}x_j \Bigr| \leqslant \sum_{j=1}^n |a_{ij} |.$$
For the reverse inequality, find $\|Ax\|$ where $x$ is the vector with a 1 in the $i$th coordinate and 0 for every other coordinate.

 

[blue_raver22]

The Maximum Norm Proof for Matrix A states that the maximum norm of a matrix A, denoted as ||A||_∞, is equal to the maximum absolute row sum of A. In other words, ||A||_∞ is equal to the maximum of the sums of the absolute values of the elements in each row of A.

To prove this, we can start by considering the definition of the maximum norm for a matrix A. This is defined as the maximum of the norms of all possible vectors x in ℝ^n, such that ||x||_∞ = 1. In other words, we are looking for the maximum value of ||Ax||_∞ over all possible unit vectors x.

We can rewrite ||Ax||_∞ as the maximum absolute row sum of A multiplied by the norm of x. This is because the norm of x is equal to 1, and the maximum absolute row sum of A is the maximum value that can be obtained by multiplying A with any unit vector x.

So, we have ||Ax||_∞ = ||A||_∞ ||x||_∞. Substituting this into the original definition, we get:

||A||_∞ = max ||Ax||_∞ / ||x||_∞

Now, let's consider the definition of the maximum absolute row sum of A. This is equal to the maximum of the sums of the absolute values of the elements in each row of A. We can rewrite this as:

max ||Ax||_∞ / ||x||_∞ = max (|a_{11}| + |a_{12}| + ... + |a_{1n}|, |a_{21}| + |a_{22}| + ... + |a_{2n}|, ..., |a_{n1}| + |a_{n2}| + ... + |a_{nn}|)

Since we are looking for the maximum value of this expression, we can simplify it further by considering the maximum value of each term in the expression. This gives us:

max ||Ax||_∞ / ||x||_∞ = max (max |a_{11}|, max |a_{12}|, ..., max |a_{1n}|, max |a_{21}|, ..., max |a_{n1}|, ..., max |a_{nn}|)

Now, we know that the maximum value of any term in