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Frobenius norm

  1. Jun 22, 2004 #1
    Hi I'm in the process of proving a matrix norm. The Frobenius norm is defined by an nxn matrix A by ||A||_F=sum[(|aij|^2)^(1/2) i=1..n,j=1..n] I'm having trouble showing ||A+B|| <= ||A|| + ||B||

    thanks for the help
     
  2. jcsd
  3. Jun 22, 2004 #2

    Hurkyl

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    What's the formula for the norm of a vector with n^2 entries?
     
  4. Jun 22, 2004 #3
    I have no idea could you enlighten me?
     
  5. Jun 23, 2004 #4

    Hurkyl

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    Er... no offense, but you can't possibly be talking about matrix norms without already having learned vector norms.

    What's the Euclidean norm of a 2-vector?
     
  6. Sep 17, 2004 #5

    mathwonk

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    the formula you gave looks wrong as well. i.e. you squared and then took square root before summinjg. so you are getting the "sum norm", whereas it seems you meant to get the "euclidean norm".

    i think hurkyl is assuming you meant the euclidean norm, and then your formula would simply be the norm of a vector in euclidean n space. the properties of this norm are probably based on some inequality they teach at the beginnig of many courses called the schwartz inequality (see chapter 0 or 1 of spivak's calculus book). it is usually proven using the quadratic formula applied to a variable t times the variables x in the vector. i.e. use the fact that a quadratic equation has a solution if and only if the discriminant b^2 -4ac is non negative.

    Actually with your formula, the sum norm, it is even easier to prove your request. indeed it seems obvious from the properties of absolute value. try it and see. of course your homework is now 3 months overdue so you are not reading this anymore.
     
  7. Dec 14, 2010 #6
    |ab|>=ab
    2|ab|>=2ab
    ||a|+|b||>=|a+b|

    Hope this helps:shy:
     
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