Moving between symmetric and

unitary matrices using Cayley transformations is not a new idea.

The capacity-achieving space time modulation signal distribution at high SNR is modelled as a set of

unitary matrices [1]: {X} = [square root of T] [{[[PHI].sub.b]}.sup.B.sub.b=1] in which each matrix satisfies [[PHI].sup.*.sub.b][[PHI].sub.b] = [I.sub.M] and all [[PHI].sub.b]'s are points on a Stiefel manifold, or the subspace [[OMEGA].sub.b] spanned by column vectors of T x M matrix is uniformly distribution in Grassmann manifold [G.sub.T,M]; that is, e [G.sub.T,M] [2].

where ([U.sub.i,1], [U.sub.i,2]) [member of] [C.sup.NxN] and ([[LAMBDA].sub.i,1], [[LAMBDA].sub.i,2]) [member of] [C.sup.MxM] are

unitary matrices. [[SIGMA].sub.i,1] = [[diag([[lambda].sub.l,l], ..., [[lambda].sub.m,1]) [0.sub.(N-M)xM]].sup.T] [member of] [C.sup.NxM] and [[SIGMA].sub.i,2] = [[diag([[lambda].sub.1,2], ..., [[lambda].sub.m,2]) [0.sub.(N-M)xM]].sup.T] [member of] [C.sup.NxM] are eigen value matrices, where [[lambda].sub.i,1] is the element of eigenvalues.

Then [P.sub.t], [Q.sub.t] are

unitary matrices. The real presentation has the following properties, which are given by Jiang and Wei [14].

By SVD, the channel matrix, H, is decomposed into a diagonal matrix, [SIGMA], and two

unitary matrices, U and V, as

Matrices [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are

unitary matrices satisfying

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the

unitary matrices of i-mode singular vectors for i = 1, 2, 3.

Yayli, Dual

unitary matrices and unit dual quaternions, Differential Geometry Dynamical Systems, 10(2008), 1-12.

U and V are real N x N

unitary matrices with small singular values.

Note that the technique used in the proof above can be extended easily to handle bigger input alphabets by using the matrices defined on Page 169 of [Paz71], and the method of simulating stochastic matrices by

unitary matrices described in [YS09b, YS10].

This third edition features new chapters on generalized eigenvectors and numerical techniques, as well as a new section on Hermitian symmetric and

unitary matrices. It offers additional exercises consisting of application, numerical, and conceptual questions.

Being subgroups of SU[(2).sub.L] x U[(1).sub.Y], they have group operations represented by 2 x 2

unitary matrices or, equivalently, by unit quaternions.