# Cayley’s parameterization of orthogonal matrices

Any orthogonal matrix^{} $O$ which does not have $-1$ as an eigenvalue^{} can be expressed as

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

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^{},

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

These two formulae are each other’s inverses^{} and set up a one-to-one correspondence between orthogonal^{} and skew-symmetric matrices.

## 0.0.1 Proof

The restriction^{} on the eigenvalues of $O$ is necessary in order for $I+O$ to be invertible.

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

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

Using the fact that the transpose^{} of a product^{} is the product of the transposes in the opposite order,

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

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 ${I}^{T}=I$.

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

Insert an identity matrix^{} between the two factors like so:

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

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

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

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

$$u=\frac{i+h}{i-h}$$ |

$$ih=\frac{u+i}{u-i},$$ |

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.

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 |
---|---|

Canonical name | CayleysParameterizationOfOrthogonalMatrices |

Date of creation | 2013-03-22 14:51:38 |

Last modified on | 2013-03-22 14:51:38 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 20 |

Author | rspuzio (6075) |

Entry type | Theorem |

Classification | msc 22E70 |

Classification | msc 15A57 |

Defines | Cayley transform |