Sometime ago, in my research, I came across a matrix map I am interested at. It is a generalisation of the complex map
defined on a complex projective plane .
Let be a field. Given a skew-symmetric matrix , i.e. , one can define the Cayley transform of , denoted by , as
for all such that is invertible.
If we were somewhat more restrictive in our assumption about , which in this case is to consider an ordered field, then is always invertible for all skew-symmetric matrix . To see this, suppose it is not, then there exists a non-zero such that , i.e. . However,
which implies . Since we are working in an ordered field, this forces , a contradiction.
It can be shown that is an orthogonal matrix, i.e. where is the identity matrix of the same size. It’s not hard to see this by direct computation:
In the proof above, we used the fact that commutes with . Moreover, because
Proposition. The map is an involution, i.e.
Example. If , then
for some satisfying . If , then
for all such that .
In Lie group and Lie algebra notation,
where is the Lie algebra of skew-symmetric matrices equipped with the usual Lie bracket, the Lie group of orthogonal matrices, and a subgroup of consisting of orthogonal matrices with determinant 1. We can then think of Cayley transformation as a map from into its Lie group .
This map is nice because it allows us parametrise rotation matrices! Generally, if one wishes to study a Lie group, one might do that by studying its Lie algebra which behaves much nicer because we can do linear algebra stuff there.
One thing I want to mention about is that it does not require infinite processes. The formula is precise and does not require approximation. Compare this with the more commonly studied exponential map , which is also a map from a Lie algebra to its Lie group :
for . In this setting, usually the underlying field is taken to be so if one is given , one needs to take a limit (which requires infinite processes) to exactly compute . It shares a similar property with the Cayley transformation: if is skew-symmetric, then is orthogonal. In some ways, exponential map is stronger because we have things like
for example, while we don’t have an analogue for this with the Cayley transformation. What I mean by that is it’s not possible to find as a function of that satisfies
for all . Nevertheless, it does not mean that the Cayley transformation is less interesting. I feel that this map needs to be more well-studied, especially because it might provide an alternate theory of connecting Lie algebras with Lie groups more algebraically.
I will discuss a generalisation of this Cayley transform to a more general geometric setting in my next post. Roughly speaking (I will make this more rigorous in the next post), given a symmetric, non-degenerate matrix , we can define a symmetric bilinear form for all . This gives a new geometry, with the case giving the Euclidean geometry back. The matrix gives relativistic geometry.
This generalised Cayley transform depends on . We will also see that a modification of being skew-symmetric and orthogonal with respect to in the next post.
Brisbane, 28 June 2019.