Кейли түрленуі: математикалық амал

Mathematical operation
In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew symmetric matrices and special orthogonal matrices. The transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators .

One can extend the mapping to complex matrices by substituting "unitary" for "orthogonal" and "skew Hermitian" for "skew symmetric", the difference being that the transpose (·T) is replaced by the conjugate transpose (·H). This is consistent with replacing the standard real inner product with the standard complex inner product. In fact, one may extend the definition further with choices of adjoint other than transpose or conjugate transpose. Formally, the definition only requires some invertibility, so one can substitute for Q any matrix M whose eigenvalues do not include −1. For example,

Note that A is skew symmetric (respectively, skew Hermitian) if and only if Q is orthogonal (respectively, unitary) with no eigenvalue −1.

An infinite dimensional version of an inner product space is a Hilbert space, and one can no longer speak of matrices. However, matrices are merely representations of linear operators, and these can be used. So, generalizing both the matrix mapping and the complex plane mapping, one may define a Cayley transform of operators. Here the domain of U, dom U, is (A+iI) dom A. See self adjoint operator for further details.