# Vector Norm and Matrix Norm

• Sep 18th 2008, 12:57 AM
tinng
Vector Norm and Matrix Norm
Let x be a m - vector, A be an m x n matrix, show that

(1) infinity norm x <= 2nd norm x <= square root (m) * infinity norm x

(2) [1/square root(n)] * infinity norm A <=2nd norm A
<=
square root (m) * infinity norm A
• Sep 18th 2008, 10:49 AM
Moo
Hello,
Quote:

Originally Posted by tinng
Let x be a m - vector, A be an m x n matrix, show that

(1) infinity norm x <= 2nd norm x <= square root (m) * infinity norm x
(2) [1/square root(n)] * infinity norm A <=2nd norm A
<=
square root (m) * infinity norm A

Get back to the definitions of the norms :

$\displaystyle \left\|x \right\|_p=\left(\sum_{i=1}^m |x_i|^p \right)^{1/p}$

$\displaystyle \left\|x \right\|_\infty=\max_{1 \le i \le m} |x_i|$

You'll have to use two inequalities :
- Triangle inequality : $\displaystyle |x+y| \le |x|+|y|$
- Cauchy-Schwarz inequality : $\displaystyle \left(\sum_{i=1}^m x_iy_i\right)^2 \le \left(\sum_{i=1}^m x_i^2 \right)\left(\sum_{i=1}^m y_i^2 \right)$