TITLE: Understanding weighted inner product and weighted norms
QUESTION [21 upvotes]: I am reading  this book where at page 27 following definitions about weighted inner product and weighted  norms are given. 
Let $M$ and $N$ be Hermitian positive definite matrices of order $m$ and $n$ respectively. The weighted inner products in $\mathbb{C}^{m}$ and $\mathbb{C}^{n}$ are 
$(x, y)_{M} = y^{*}Mx$ , $x, y \in \mathbb{C}^{m}$ and $(x, y)_{N} = y^{*}Nx$ , $x, y \in \mathbb{C}^{n}$ ....$(1)$
The definitions of weighted vector norms are
$\|x\|_{M} = (x, x)^\frac{1}{2}_{M} = (x^{*}Mx)^\frac{1}{2} = \|M^\frac{1}{2} x\|_{2}$, $x\in\mathbb{C}^{m}$ ....$(2)$
$\|x\|_{N} = (x, x)^\frac{1}{2}_{N} = (x^{*}Nx)^\frac{1}{2} = \|N^\frac{1}{2} x\|_{2}$, $x\in\mathbb{C}^{n}$ ....$(3)$
The definitions of weighted matrix norm are
$\|A\|_{MN} = \max_{\|x\|_{N} = 1}{\|Ax\|_{M}},\;  x \in\mathbb{C}^n and ~~A\in \mathbb{C}^{m\times n}$
$\|B\|_{NM} = \max_{\|x\|_{M} = 1}{\|Bx\|_{N}},\;  x \in\mathbb{C}^n and ~~B\in \mathbb{C}^{n\times m}$
Such a norm is sometimes called an operator norm subordinate to vector norm. It is easy to verify that
$\|A\|_{MN} =  \|M^\frac{1}{2} A  N^\frac{-1}{2} \|_{2}$ ....$(4)$
$\|B\|_{NM} =  \|N^\frac{1}{2} B  M^\frac{-1}{2} \|_{2}$ ....$(5)$
Could anybody explain me about the significance of weighted norms? Why we need weighted norm? In $(2)$ how we got $\|M^\frac{1}{2} x\|_{2}$? How could we find square root of matrix $M$?  How did we got equation $(4)$ and $(5)$. 
I would be very much thankful for the help and suggestions.

REPLY [19 votes]: The above answers are perfectly nice. I just want to point out another example: energy norms.
I don't know how familiar you are with differential equations and/or calculus of variations, but I'll give it a try anyway.
Consider the following integral:
$$
E(v) = \frac{1}{2}\int_\Omega |\nabla v|^2 dx
$$
where $\Omega$ is a nice (say with a smooth boundary, no corners nor spikes) bounded domain in $\mathbb{R}^n$. This in many application represents the internal energy of a system in a configuration given by the function $v$. For instance, if $v$ is the displacement from a reference configuration, $E(v)$ represents the elastic energy of the system (assuming linear elasticity).
The above integral can be rewritten as
$$
E(v) = a(v,v)
$$
with
$$
a(u,v) = \frac{1}{2}\int_\Omega \nabla u\cdot\nabla v dx
$$
Now, suppose we have a finite dimensional representation of the function $v$ (if you know finite elements you know where I'm heading). This means
$$
v(x) = \sum_{i=1}^n v_i \varphi_i(x),
$$
where all the $\varphi_i(x)$ are fixed and known a priori.
If you plug this expression inside the definition of $E$ you get (being careful not to mess up the indices)
$$
E(v) = \frac{1}{2}\int_\Omega \nabla\left(\sum_{j=1}^n v_j \varphi_j(x)\right)\cdot\nabla\left(\sum_{i=1}^n v_i \varphi_i(x)\right)dx\\
= \cdots = \sum_{i=1}^n\sum_{j=1}^n v_iv_j\frac{1}{2}\int_\Omega \nabla \varphi_j \cdot \nabla \varphi_i dx
$$
Now let $\underline{v}$ be the vector of the coefficients $v_i$ and $A$ the matrix whose entries are
$$
a_{ij} = a(\varphi_j,\varphi_i) = \frac{1}{2}\int_\Omega \nabla \varphi_j\cdot\nabla\varphi_i dx
$$
Under certain assumptions on $v$ (for instance, $v=0$ on $\partial\Omega$, the boundary of $\Omega$), it can be shown that this is indeed a positive definite matrix.
Now, if the system is in a configuration described by $v$ and $v$ is expressed as above, then the energy of the system is given by
$$
E(v) = a(v,v) = \underline{v}^tA\underline{v}
$$
which is precisely a weighted norm of $\underline{v}$ (squared). Here the matrix is not exactly a weight, bur rather it encodes the physics of the phenomenon. It is possible to show that, if $v$ is expanded as before, and you pick your basis functions $\varphi_i$ in such a way that
$$
\int_\Omega \varphi_i\varphi_j dx = 
\begin{cases}
0\mbox{ if }i\neq j\\
1\mbox{ if }i=j,
\end{cases}
$$
then the standard Eucledian norm of $\underline{v}$ is corresponds to the value of the integral
$$
I(v) = \int_\Omega v^2 dx
$$
which is an important norm of $v$, but it measures a different energy of the system.