Cayley’s parameterization of orthogonal matrices
It is a matter of simple computation why these formulae are correct. Suppose that is skew-symmetric. Then
Using the fact that the transpose of a sum is the sum of transposes and the transpose of an inverse is the inverse of the transpose,
By the definition of skew-symmetry, and ,
Finally, since and commute, we may switch the order of the second and third factors:
Then the first two factors and the last two factors cancel, showing that .
Next, we verify that the second formula is indeed the inverse of the first formula. Multiplying by on both sides,
Expanding this and moving terms from one side of the equation to the other,
Multiplying both sides by , we obtain the desired formula:
Finally, one can show that, if is orthogonal, then is skew-symmetric using the same sort of computation that was used to show the converse:
Using the facts about transposes of sums, products, and inverses,
Since is orthogonal, . As usual .
Insert an identity matrix between the two factors like so:
Replace the identity matrix with :
Absorbing the and the into the factors,
and commute, and consequently
0.0.2 Cayley Transform
The relation between and which is set up by the formulas
is sometimes known as the Cayley transform. Note that the proof that these two formulas are each other’s inverses did not require to be skew-symmetric or to be orthogonal. Hence, the Cayley transform is defined for all matrices such that is not an eigenvalue of . (Recall that this condition is necessary to insure that is invertible.
The Cayley parameterization can be generalized to unitary transforms. Namely, if is a unitary matrix, then is the Cayley transform of a skew-Hermitean matrix . Since a skew-Hermitean matrix can be written as times a Hermitean matrix, the Cayley transform is often written as follows when dealing with unitary matrices:
A special case of this worth pointing out is the case of one-dimensional unitary matrices. The sole entry of a one dimensional unitary matrix must have modulus 1 and the sole entry of a one-dimensional Hermitean matrix must be real. In that case, the Cayley transform reduces to
which is a fractional linear transform that maps the unit circle to the real axis.
The Cayley parameterization can be generalized to the case of a general inner product with arbitrary signature (see Sylvester’s law for the definition of signature — Cayley and Sylvester were the best of friends). We simply need to define the transpose of a matrix by the condition for all vectors and . In particular, this allows one to parameterize pseudo-orthogonal matrices such as Lorentz transformations using a Cayley parameterization. Likewise, given a conjugate linear inner product on a complex vector space, one has a Cayley parameterization of the unitary (or pseudo-unitary) transforms which preserve the product.
In conclusion, it might be worth pointing out that the Cayley transform generalizes to the case of infinite dimensions, if one replaces matrices with operators on a Hilbert space. In particular, it is useful because unitary and orthogonal operators are bounded whereas Hermitean and skew-symmetric operators may or may not be bounded. For instance, it is often easier to obtain the spectral decomposition of a Hermitean operator or study symmetric extensions of a symmetric operator by first performing a Cayley transform and dealing with the resulting bounded operator.
|Title||Cayley’s parameterization of orthogonal matrices|
|Date of creation||2013-03-22 14:51:38|
|Last modified on||2013-03-22 14:51:38|
|Last modified by||rspuzio (6075)|