# Cayley’s parameterization of orthogonal matrices

 $O=(I+A)(I-A)^{-1}$

for some suitable skew-symmetric matrix $A$. Conversely, any skew-symmetric matrix $A$ can be expressed in terms of a suitable orthogonal matrix $O$ by a similar  formula   ,

 $A=(O+I)^{-1}(O-I).$

## 0.0.1 Proof

It is a matter of simple computation why these formulae are correct. Suppose that $A$ is skew-symmetric. Then

 $O^{T}O=\left((I+A)(I-A)^{-1}\right)^{T}(I+A)(I-A)^{-1}$
 $=((I-A)^{-1})^{T}(I+A)^{T}(I+A)(I-A)^{-1}$

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,

 $=(I^{T}-A^{T})^{-1}(I^{T}+A^{T})(I+A)(I-A)^{-1}$

By the definition of skew-symmetry, $A^{T}=-A$ and $I^{T}=I$,

 $=(I+A)^{-1}(I-A)(I+A)(I-A)^{-1}$

Finally, since $I+A$ and $I-A$ commute, we may switch the order of the second and third factors:

 $=(I+A)^{-1}(I+A)(I-A)(I-A)^{-1}$

Then the first two factors and the last two factors cancel, showing that $O^{T}O=I$.

Next, we verify that the second formula is indeed the inverse of the first formula. Multiplying by $I-A$ on both sides,

 $O(I-A)=I+A$

Expanding this and moving terms from one side of the equation to the other,

 $O-I=A+OA$

Factoring,

 $O-I=(I+O)A$

Multiplying both sides by $(I+O)^{-1}$, we obtain the desired formula:

 $(O+I)^{-1}(O-I)=A$

Finally, one can show that, if $O$ is orthogonal, then $A$ is skew-symmetric using the same sort of computation that was used to show the converse:

 $A^{T}=\left((O+I)^{-1}(O-I)\right)^{T}$

Using the facts about transposes of sums, products, and inverses,

 $=(O^{T}-I^{T})(O^{T}+I^{T})^{-1}$

Since $O$ is orthogonal, $O^{T}=O^{-1}$. As usual $I^{T}=I$.

 $=(O^{-1}-I)(O^{-1}+I)^{-1}$
 $=(O^{-1}-I)I(O^{-1}+I)^{-1}$

Replace the identity matrix with $OO^{-1}$:

 $=(O^{-1}-I)OO^{-1}(O^{-1}+I)^{-1}$

Absorbing  the $O$ and the $O^{-1}$ into the factors,

 $=(I-O)(I+O)^{-1}=-(O-I)(O+I)^{-1}.$

Since

 $O(O+I)^{-1}=((O+I)O^{T})^{-1}=(I+O^{T})^{-1}=(O^{T}(O+I))^{-1}=(O+I)^{-1}O,$

$(O-I)$ and $(O+I)^{-1}$ commute, and consequently

 $=-(O+I)^{-1}(O-I)=-A.$

## 0.0.2 Cayley Transform

The relation   between $A$ and $O$ which is set up by the formulas

 $O=(I+A)(I-A)^{-1}$

and

 $A=(O+I)^{-1}(O-I)$

is sometimes known as the Cayley transform. Note that the proof that these two formulas are each other’s inverses did not require $A$ to be skew-symmetric or $O$ to be orthogonal. Hence, the Cayley transform is defined for all matrices such that $-1$ is not an eigenvalue of $O$. (Recall that this condition is necessary to insure that $O+I$ is invertible.

## 0.0.3 Generalizations

The Cayley parameterization can be generalized to unitary   transforms. Namely, if $U$ is a unitary matrix, then $U$ is the Cayley transform of a skew-Hermitean matrix $A$. Since a skew-Hermitean matrix can be written as $i$ times a Hermitean matrix, the Cayley transform is often written as follows when dealing with unitary matrices:

 $U=(iI+H)(iI-H)^{-1}$
 $iH=(U+I)^{-1}(U-I)$

where $H$ is Hermitean. The proof in this case is substantially the same as was presented above; all one has to do is replace matrix transposition  with Hermitean conjugation  .

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

 $u={i+h\over i-h}$
 $ih={u+i\over u-i},$

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 $M$ by the condition $(M^{T}u)\cdot v=u\cdot(Mv)$ for all vectors $u$ and $v$. 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.

Title Cayley’s parameterization of orthogonal matrices CayleysParameterizationOfOrthogonalMatrices 2013-03-22 14:51:38 2013-03-22 14:51:38 rspuzio (6075) rspuzio (6075) 20 rspuzio (6075) Theorem msc 22E70 msc 15A57 Cayley transform