Results 1 to 4 of 4

Math Help - Von Neumann inequality

  1. #1
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    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 \|A\| where A=\frac{1}{2}I_3?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Quote Originally Posted by Drexel28 View Post
    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 \|A\| where A=\frac{1}{2}I_3?
    For your example the usual definition works fine since \Vert Av \Vert = 1/2 for all v\in \mathbb{S} ^2. For diagonal matices (or any matrix for which A^*A is simple enough) you could use \Vert A \Vert = \sqrt{ \lambda _{max} (A^*A) } where \lambda _{max} (A^*A) is the largest eigenvaule of A^*A and A^* means the adjoint of A.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hello,

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

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

    where \|(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) :

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

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

    \|A\|_2=\sqrt{\rho(A'A)}, where \rho(M)=\max ~|\lambda_i| ( \lambda_i are the eigenvalues of M)
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member Rebesques's Avatar
    Joined
    Jul 2005
    From
    At my house.
    Posts
    542
    Thanks
    11
    For example what is \|A\| where 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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Projections on Neumann algebra
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: April 22nd 2011, 01:43 PM
  2. [SOLVED] Neumann Boundary Conditions
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: November 21st 2010, 01:55 PM
  3. [SOLVED] von Neumann Ordinals
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: March 19th 2010, 12:08 PM
  4. PDE - Wave eqn with Neumann
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: May 12th 2009, 11:02 AM
  5. Von Neumann Numbers
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: October 3rd 2006, 08:40 AM

Search Tags


/mathhelpforum @mathhelpforum