They've actually measured the diameter of a black hole.
The radius may be undefined. At any rate, if the radius IS defined, then no, it doesn't have infinite density.
I'm not sure how accurate
black hole theory is (I think at best it's probably incomplete since we lack a theory of quantum gravity), but to the extent we think we understand
them, we need to define the different types of radii. I'm not going to discuss them all because that will just unnecessarily confuse the issue.
That article is about an event horizon, which is known also as a Schwarzchild radius. That type of radius is not relevant to the rotating matter
question, as there is no matter rotating at that diameter in the absence of an accretion disk, according to black hole theory.
The horizon is not a physical surface, merely a conceptual one, and although it marks the point of no return for material plummeting toward the
singularity, relativity says that nothing special happens there
In fact the theory states it's only apparent to an external observer (like us measuring it in that article) but if you flew into that black hole in a
space ship, you'd fly right past the event horizon without even realizing anything was there, because nothing is there; there's no matter to rotate
(The radius measurement article references a supermassive black hole, which can have quite large event horizons). Our own supermassive black hole at
the center of the Milky Way has had nothing at the event horizon for decades, though recently we observed a gas cloud approaching it and we are
looking forward to observing what happens when it "feeds" upon the gas cloud, such as whether we will see Synchrotron radiation or whatever. I'm
hoping we can measure or at least estimate the speed of the matter falling in, but the point is, previously, there hasn't been any matter falling in
since we've been observing it.
The other relevant radius is of the collapsed matter inside the black hole. If it's a non rotating black hole, it's a point with no volume (you
could call it a radius of zero), or if it's a rotating black hole, it's a 2-dimensional disc with no volume which would have a radius far smaller
than that of the event horizon.
Black Hole Singularity
At the center of a black hole as described by general relativity lies a gravitational singularity, a region where the spacetime curvature becomes
infinite. For a non-rotating black hole, this region takes the shape of a single point and for a rotating black hole, it is smeared out to form a ring
singularity lying in the plane of rotation. In both cases, the singular region has zero volume. It can also be shown that the singular region contains
all the mass of the black hole solution. The singular region can thus be thought of as having infinite density.
The problem is, our theories "break down" inside the event horizon, meaning what we define as "space-time" doesn't exist inside the event
horizon, so, if there's no time, so how can you calculate a velocity? Even at the event horizon, time appears to crawl to a standstill to outside
observers leading to the term "frozen stars" but that gets into the other radii not really relevant to the rotating matter question in this thread.
The appearance of singularities in general relativity is commonly perceived as signaling the breakdown of the theory. This breakdown, however,
is expected; it occurs in a situation where quantum effects should describe these actions, due to the extremely high density and therefore particle
interactions. To date, it has not been possible to combine quantum and gravitational effects into a single theory, although there exist attempts to
formulate such a theory of quantum gravity. It is generally expected that such a theory will not feature any singularities.
There may be a way to do it using a theory of quantum gravity or some other theory like an unproven hypothetical string theory, but currently I don't
know how to do it and even if I did, we don't know of any way to make observations inside the event horizon to determine if the math is right or not.