Here’s a script which might help if you’d like to try Makie. Makie takes a while to load, but by gosh it is slickly interactive!
Before going on, I’d like to note that visualizing this tensor K of yours may be tricky because it is not symmetric. You tackle that in your example by taking svd(K) and considering only the singular values, but it’s worth thinking about whether this is what you want for your application. (Are the matrices U and V also important for you? They will define two different rotated frames)
With all those questions in mind, in the following I’ve cooked up a visualization which measures the length of K*v for each vector v on the unit sphere, and also color coded the output vector K*v in the color. Maybe this is not at all what you want though! (If you could describe your application briefly perhaps we could come up with a more appropriate visualization.)
using Makie
using LinearAlgebra
# Utility to split an array-of-length-3-arrays into x,y,z components
split_coords(ps) = getindex.(ps,1), getindex.(ps,2), getindex.(ps,3)
# Utility to convert vectors to colors
to_rgb(v) = RGBf0(v[1], v[2], v[3])
# Convert spherical coordinates to Cartesian coordinates.
# Note that θ,ϕ are named according to the "physicist convention"
# Note that using Vec3f0 here is efficient, but you could just use a
# normal array and everything would still work.
spherical_to_cartesian(r,θ,ϕ) = Vec3f0(r*sin(θ)*cos(ϕ), r*sin(θ)*sin(ϕ), r*cos(θ))
θ = range(0,pi,length=20)
ϕ = permutedims(range(0,2pi,length=30))
K =[ 0.070087 -0.0386396 0.00209088;
-0.0441344 0.105735 0.00012994;
-0.00553273 -0.00292807 0.0896236]
# A parametric unit sphere
v = spherical_to_cartesian.(1.0, θ, ϕ)
# One way of "seeing how K acts in each direction" is by taking the norm of
# K * v and using this as the radius in terms of the original angular
# coordinates.
r = norm.(Ref(K) .* v)
ps = spherical_to_cartesian.(r, θ, ϕ)
# Another way is by seeing how it distorts the unit sphere by computing K*v for
# every point v on the sphere:
# ps = Ref(K) .* v
# Ultimately both of the above techniques only present part of the information
# about K in the position because K contains more degrees of freedom than we
# can represent on an ellipse.
x,y,z = split_coords(ps)
scene = surface(x,y,z, color=to_rgb.(v))
x,y,z = split_coords(1.003.*ps) # Scaling hack to avoid some z-fighting
wireframe!(scene, x,y,z)
# Makie.save("tensor.png", scene)
