Final answer:
A non-symmetric tensor in 3D can be visualized as a skewed ellipsoid, which requires transforming the standard Cartesian coordinate system to reflect the tensor's complex nature.
In higher dimensions, such tensors might be represented through projections or reductions to concepts that fit within our three-dimensional understanding.
Step-by-step explanation:
To visualize a non-symmetric tensor in 3D, we cannot rely solely on the concept of a spheroid as we do for symmetric tensors.
A non-symmetric tensor in three dimensions can be represented graphically by imagining an ellipsoid that does not necessarily align with the coordinate axes or might be skewed.
This ellipsoid would have three distinct axes, but unlike the symmetric case, the axes do not represent the principal directions of the tensor. A non-symmetric tensor possesses not only magnitude in different directions but also has mixed components that represent shearing effects.
In three-dimensional space, each point can be described by a Cartesian coordinate system, where the x, y, and z axes are orthogonal. In such a system, the unit vectors i, j and k are employed to define orientation, with a common right-handed arrangement as the standard.
However, to capture the complexity of a non-symmetric tensor, one might have to apply a transformation to these axes, which would result in a skewed coordinate frame, more aptly reflecting the nature of the non-symmetric tensor.
Visualizing tensors in higher dimensions, beyond 3D, arises in contexts such as the theory of relativity where spacetime is four-dimensional.
This complexity can be approached by reducing dimensions, like using a light cone to represent the 4th dimension of time in a 3D space, or representing higher-dimensional objects like the 6D hypercube in theoretical frameworks.
These models rely on projecting or reducing higher dimensions into a form that can be understood within our three-dimensional perception.