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Thread: Using CS inequality

  1. #1
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    Using CS inequality

    I need to show that if $\displaystyle \bf{M}$ is an $\displaystyle m \times n$ matrix, then $\displaystyle ||\bf{M}(\bf{h})||\leq \alpha ||\bf{h}||$ for some $\displaystyle \alpha$ by estimating the entries of $\displaystyle \bf{M}$. The hint I am given is that $\displaystyle \bf{M}(\bf{h}) = \bf{y} = (y_1, \ldots, y_m)$ where $\displaystyle y_i=\bf{m}_i \cdot \bf{h}$ for some $\displaystyle \bf{m}_i \in R^n$ and apply the CS inequality.

    I have not done multivariable calc in a VERY long time, and I just can't seem to jumpstart my brain to attack this problem.
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  2. #2
    Super Member girdav's Avatar
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    If we denote by $\displaystyle (a_{k,j})_{1\leq k\leq m,1\leq j\leq n}$ the coefficients of the matrix $\displaystyle M$, we have to notice that $\displaystyle y_i =\sum_{j=1}^na_{ij}h_j$. Now, the question is: what is the vector $\displaystyle m_i$ and what does the Cauchy-Schwarz inequality give?
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