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Math Help - Vector Norm and Matrix Norm

  1. #1
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    Question 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
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    Hello,
    Quote Originally Posted by tinng View Post
    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 :

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

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


    You'll have to use two inequalities :
    - Triangle inequality : |x+y| \le |x|+|y|
    - Cauchy-Schwarz inequality : \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)
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