Cayley Transform - Matrix Map

Matrix Map

Among n×n square matrices over the reals, with I the identity matrix, let A be any skew-symmetric matrix (so that AT = −A). Then I + A is invertible, and the Cayley transform

produces an orthogonal matrix, Q (so that QTQ = I). The matrix multiplication in the definition of Q above is commutative, so Q can be alternatively defined as . In fact, Q must have determinant +1, so is special orthogonal. Conversely, let Q be any orthogonal matrix which does not have −1 as an eigenvalue; then

is a skew-symmetric matrix. The condition on Q automatically excludes matrices with determinant −1, but also excludes certain special orthogonal matrices. Some authors use a superscript "c" to denote this transform, writing Q = Ac and A = Qc.

This version of the Cayley transform is its own functional inverse, so that A = (Ac)c and Q = (Qc)c. A slightly different form is also seen (Golub & Van Loan 1996), requiring different mappings in each direction (and dropping the superscript notation):

The mappings may also be written with the order of the factors reversed (Courant & Hilbert 1989, Ch.VII, §7.2); however, A always commutes with (μI ± A)−1, so the reordering does not affect the definition.