Conference rusure::math

1500.0. "Normed Vector Spaces talk" by STAR::ABBASI () Tue Oct 15 1991 06:35

    give a valid norm for a vector space V over real number field, where
    the elements of V are m,n matrices.
    
    a valid norm is a function (N) that maps the element in V into R (real
    field) such that , for x,y in V , and 'a' is a scalar in R :
    
    1. N(x) >= 0
    2. N(x) = 0 iff x=0
    3. N(ax) = |a| N(x)
    4. N(x+y) <= N(x) + N(y)
    
    one example i see now is N(x) to be the absolute value of determinant. but
    i have to assume that n=m which is not general.
    
    so what i did, is say that n,m matrix is like q sized vector where
    q= m*n . (ie. traverse matrix left to right, top to bottom) and the norm 
    in this case is 
    
                             
            -   m*n                   -  1/p
            |  -----          p        |
     N(x) = |  \        | x  |         |
      p     |  /           k           |
            |  -----                   |
            -   k=1                   -
    
    this, i can show easly that it meets the above 4 conditions for valid 
    norm for any p value 1..oo
    
    note that for p=2 we get the "normal" length of vector. for p=oo
    things looks funny, like a circle becomes a square etc.. for p=0
    undefined. and p not negative. 
    
    but im sure that there is some function that can be applied
    to an m,n matrix that return an element in R s.t. the above conditions
    are met without me "looking" at the matrix as a vector.
    i cant see it know. any one know such a function?
    
    i've allready submitted the homework, will find out next week what is
    the answer, and will post teacher's answer if different than anyones,
    meanwhile you can have fun with this.
    
    /nasser

    
    i think i just saw one function
    
    N(x) = total sum of absolute value of elements of matrix.
    
    since we add matrices component wise, this looks ok.
    
    /nasser
    

    ref .-1
    Nasser, this function is the same as .0 when p=1 . wake up.
    

    Your .0 formulation is close to one of the standard normalizations
    presented in the vector spaces courses I took thirty years ago.  I
    predict you will get an acceptable grade on your homework.
    
    Dick