  This article first introduces the notion of weighted singular value
decomposition (WSVD) of a tensor via the Einstein product. The WSVD is then
used to compute the weighted Moore-Penrose inverse of an arbitrary-order
tensor. We then define the notions of weighted normal tensor for an even-order
square tensor and weighted tensor norm. Finally, we apply these to study the
theory of numerical range for the weighted Moore-Penrose inverse of an
even-order square tensor and exploit its several properties. We also obtain a
few new results in the matrix setting that generalizes some of the existing
results as particular cases.
