NN - norm notation I need brief introduction and name/term  of the RHS of following:
$|| Q_{t}^{1/2}(w-w_t)||_{2}^{2} = ||(w-w_t)||_{Q_{t}}^{2}$
the $Q_{t}$ is a diagonal matrix, the $w$ and $w_t$ is a vector. What I find a bit perplexing is the $||...||_2$ is exchanged for $||...||_{Q_{t}}$. Does that simply mean that Euclidean norm is performed on the weighted difference $(w-w_t)$, where the weights are $Q_t$ ? Does this have a proper name ?
 A: Whenever we have a positive definite symmetric $n \times n$ matrix $Q$, we can define an inner product on $\mathbb R^n$ by
$$\langle v,w\rangle_Q = v^tQw$$
where positive definiteness of $Q$ implies $\langle v,v\rangle_Q > 0$ for $v \neq 0$. Each such $Q$ gives $\mathbb R^n$ the structure of an inner product space, and we can define the norm $\Vert v\Vert_Q = \sqrt{\langle v,v\rangle_Q}$. For $Q=I_n$ we obtain the usual Euclidean norm which you denoted $\Vert\cdot\Vert_2$.
One could call it the norm with respect to $Q$, as here: norm with respect to matrix. Not matrix norm, because that would be the norm of $Q$ itself as a linear operator.

Explicitly, if $Q = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)$, the norm of $v = (v_i)$ is $$\sqrt{\lambda_1v_1^2+\cdots+\lambda_n v_n^2}$$
so that the norm is indeed a weighted version of the Euclidean norm, with weights the diagonal entries of $Q$. But this is not the Euclidean norm applied to the vector weighed with weights $\lambda_i$, rather, their square roots:
$$\Vert v\Vert_Q = \sqrt{(\sqrt{\lambda_1}v_1)^2 + \cdots + (\sqrt{\lambda_n}v_n)^2}$$

For general $Q$, note that $$\Vert v\Vert_Q^2 = v^t Q v = v^t (Q^{1/2})^tQ^{1/2}v = \Vert Q^{1/2}v\Vert_2^2$$
where $Q^{1/2}$ is the unique symmetric positive definite square root of $Q$. (Obtained e.g. by diagonalizing $Q$ with an orthogonal matrix and taking the square roots of its (strictly positive) eigenvalues.)
