# Von Neumann inequality

• Jan 11th 2010, 01:38 PM
Drexel28
Von Neumann inequality
Hello friends.

A friend of mine is doing research with a professor on the 3 by 3 multivariable case of the Von Neumann inequality. I am helping him out a little bit, but we are running into some techincal problems. For example, what is the quickest way to calculate the operator norm of a contraction matrix? For example what is $\displaystyle \|A\|$ where $\displaystyle A=\frac{1}{2}I_3$?
• Jan 11th 2010, 02:53 PM
Jose27
Quote:

Originally Posted by Drexel28
Hello friends.

A friend of mine is doing research with a professor on the 3 by 3 multivariable case of the Von Neumann inequality. I am helping him out a little bit, but we are running into some techincal problems. For example, what is the quickest way to calculate the operator norm of a contraction matrix? For example what is $\displaystyle \|A\|$ where $\displaystyle A=\frac{1}{2}I_3$?

For your example the usual definition works fine since $\displaystyle \Vert Av \Vert = 1/2$ for all $\displaystyle v\in \mathbb{S} ^2$. For diagonal matices (or any matrix for which $\displaystyle A^*A$ is simple enough) you could use $\displaystyle \Vert A \Vert = \sqrt{ \lambda _{max} (A^*A) }$ where $\displaystyle \lambda _{max} (A^*A)$ is the largest eigenvaule of $\displaystyle A^*A$ and $\displaystyle A^*$ means the adjoint of $\displaystyle A$.
• Jan 11th 2010, 02:54 PM
Moo
Hello,

Like for the vectors, there are many norms for a matrix...

In general, we define $\displaystyle \|A\|_p=\max_{\|x\|_p\neq 0} ~\frac{\|Ax\|_p}{\|x\|_p}=\max_{\|x\|_p=1} ~\|Ax\|_p$

where $\displaystyle \|(x_1,\dots,x_n)\|_p=\left(\sum_{i=1}^n |x_i|^p\right)^{\frac 1p}$

This is easy to deal with if A is nice... Otherwise, there are a few results (that can be proved) :

$\displaystyle \|A\|_\infty=\max_{1\leq i\leq n} \sum_{j=1}^n |a_{ij}|$

$\displaystyle \|A\|_1=\max_{1\leq j\leq n} \sum_{i=1}^n |a_{ij}|$

$\displaystyle \|A\|_2=\sqrt{\rho(A'A)}$, where $\displaystyle \rho(M)=\max ~|\lambda_i|$ ($\displaystyle \lambda_i$ are the eigenvalues of M)
• Jan 13th 2010, 12:41 PM
Rebesques
Quote:

For example what is $\displaystyle \|A\|$ where $\displaystyle A=\frac{1}{2}I_3$?

1/2. I took a uni course once, where the professor would throw students out of the classroom for asking something like this. (Speechless)