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Thread: Euclidian norm

  1. #1
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    Euclidian norm

    We have the Euclidean norm (or $\displaystyle l_2-norm$) for a vector on the vector space $\displaystyle R^m$ ; $\displaystyle \parallel{v}\parallel_2 = \sqrt{\sum_{j=1}^m\mid{v_j}\mid^2}$. Given this vector norm on $\displaystyle R^m$ the induced matrix norm is; $\displaystyle \parallel{A}\parallel_2 = \sqrt{max. eigenvalue\ of\ A^T{A}}$. Let $\displaystyle \\ U\ \in\ R^{m\times{m}}$ be orthogonal ($\displaystyle U^T{U}=I).$ $\displaystyle \\ $ I want to show that 1) $\displaystyle \parallel{Uv}\parallel{{_2^2}}=\parallel{v}\parall el{_2^2} $ for all $\displaystyle v\in\ R^m$ and 2) that $\displaystyle \parallel{UA}\parallel_2 = \parallel{AU}\parallel_2$ for any $\displaystyle A \in\ R^{m\times{m}}$
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  2. #2
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    Quote Originally Posted by bram kierkels View Post
    We have the Euclidean norm (or $\displaystyle l_2-norm$) for a vector on the vector space $\displaystyle R^m$ ; $\displaystyle \parallel{v}\parallel_2 = \sqrt{\sum_{j=1}^m\mid{v_j}\mid^2}$. Given this vector norm on $\displaystyle R^m$ the induced matrix norm is; $\displaystyle \parallel{A}\parallel_2 = \sqrt{max. eigenvalue\ of\ A^T{A}}$. Let $\displaystyle \\ U\ \in\ R^{m\times{m}}$ be orthogonal ($\displaystyle U^T{U}=I).$ $\displaystyle \\ $ I want to show that 1) $\displaystyle \parallel{Uv}\parallel{{_2^2}}=\parallel{v}\parall el{_2^2} $ for all $\displaystyle v\in\ R^m$ and 2) that $\displaystyle \parallel{UA}\parallel_2 = \parallel{AU}\parallel_2$ for any $\displaystyle A \in\ R^{m\times{m}}$
    the second part of your problem is trivial because $\displaystyle (UA)^TUA=A^TU^TUA=A^TA$ and $\displaystyle (AU)^TAU=U^TA^TAU=U^{-1}A^TAU$ and we know that similar matrices have the same eigenvalues.

    for the first part, suppose $\displaystyle e_j, \ 1 \leq j \leq m,$ is the $\displaystyle j$-th column of $\displaystyle U.$ since $\displaystyle U^{-1}=U^T,$ we have $\displaystyle e_i \cdot e_j = \delta_{ij},$ where $\displaystyle \delta_{ij}$ is the Kronecker's delta. now suppose $\displaystyle v=\begin{bmatrix}v_1 & v_2 & . & . & . & v_m \end{bmatrix}^T.$ then:

    $\displaystyle ||Uv||_2^2=Uv \cdot Uv=\left(\sum_{i=1}^m v_ie_i \right) \cdot \left(\sum_{j=1}^m v_je_j \right)=\sum_{1\leq i,j \leq m}v_iv_j \delta_{ij}=\sum_{j=1}^mv_j^2=||v||_2^2.$
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    the second part of your problem is trivial because $\displaystyle (UA)^TUA=A^TU^TUA=A^TA$ and $\displaystyle (AU)^TAU=U^TA^TAU=U^{-1}A^TAU$ and we know that similar matrices have the same eigenvalues.

    for the first part, suppose $\displaystyle e_j, \ 1 \leq j \leq m,$ is the $\displaystyle j$-th column of $\displaystyle U.$ since $\displaystyle U^{-1}=U^T,$ we have $\displaystyle e_i \cdot e_j = \delta_{ij},$ where $\displaystyle \delta_{ij}$ is the Kronecker's delta. now suppose $\displaystyle v=\begin{bmatrix}v_1 & v_2 & . & . & . & v_m \end{bmatrix}^T.$ then:

    $\displaystyle ||Uv||_2^2=Uv \cdot Uv=\left(\sum_{i=1}^m v_ie_i \right) \cdot \left(\sum_{j=1}^m v_je_j \right)=\sum_{1\leq i,j \leq m}v_iv_j \delta_{ij}=\sum_{j=1}^mv_j^2=||v||_2^2.$
    .

    Thanks a lot,
    I have one more question about this norm;
    Given a vector norm $\displaystyle ||\cdot||$ on $\displaystyle R^m$ the induced matrix norm for m x m matrices A is defined by
    $\displaystyle ||A||\ =\ max_{v\neq{0}}\frac{||Av||}{||v||}$
    That is , $\displaystyle ||A||$ is the smallest number $\displaystyle \alpha$ such that $\displaystyle ||Av||\leq\alpha||v||\ \forall\ v \in R^m$
    So given the Euclidian-norm for a vector as written in the first question why is the induced matrix norm $\displaystyle ||A||_2=\sqrt{max.\ eigenvalue A^TA}$
    In my book it is given but I donīt see it directly, i think it has something to do with the fact that $\displaystyle A^TA$ is symmetric.
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