can any one show the steps how I convert the summation to norm as below: $$ \sum_{j=1}^{M'} \big(\lambda_j (\mathbf{L}\mathbf{x})^2_j\big) = \|\mathbf{L}_\lambda \mathbf{x}\|^2, \qquad\text{where } \mathbf{L}_\lambda = \mathbf{\Lambda}^{1/2}\mathbf{L} \text{ and } \mathbf{\Lambda} = \mathrm{diag}(\lambda). $$ where $L$ is a matrix, and $x$ is a vector.
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1$\begingroup$ you can also do it the other way around; just use the definition of the vector 2-norm and that of the matrix-vector product to arrive at the sum on the LHS. this does not quite fall under "computational science" ... is it perhaps a homework problem? if so you should label it as such. $\endgroup$ – GoHokies May 18 '17 at 13:48
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Define $u_j = (L\boldsymbol{x})_j$. Then we can do the following:
\begin{align} \sum_{j=1}^{n} \lambda_j u_j^2 &= \sum_{j=1}^{n} (\lambda_j^{1/2} u_j)^2 \\ &= (\Lambda^{1/2}\boldsymbol{u})^{T}(\Lambda^{1/2}\boldsymbol{u}) \\ &= \left\lVert \Lambda^{1/2}\boldsymbol{u} \right\rVert^2 \\ &= \left\lVert \Lambda^{1/2}L\boldsymbol{x} \right\rVert^2 \\ &= \left\lVert L_{\lambda}\boldsymbol{x} \right\rVert^2 \end{align}
where, as you put it, $L_{\lambda} = \Lambda^{1/2}L$.
