where [[??].sub.i,m] and [mathematical expression not reproducible] are unitary matrices
satisfying [mathematical expression not reproducible] and [mathematical expression not reproducible].
where P and Q are p x p unitary matrices
, [SIGMA] is a diagonal p x p matrix with nonnegative real numbers, and [Q.sup.*] is the complex conjugate transpose of Q.
where the time-independent matrices [K.sub.n], n [member of] N, are arbitrary 2 x 2 unitary matrices
; that is, they satisfy the relations
Arguably no other class of matrices has enjoyed such success, with one exception, unitary matrices
. Unitary matrices
share many of the same properties as symmetric ones: normality, localized eigenvalues, and special condensed forms, which allow to represent symmetric and unitary matrices
with two vectors.
[[U.sub.1,k] [U.sub.2,k]] and [[V.sub.1,k] [V.sub.2,k]] are unitary matrices
where U and V are N x N unitary matrices
(i.e., [UU.sup.T] = [U.sup.T]U = I), and [summation] is an N x N diagonal matrix containing the singular values.
The optimal packing method of searching the orthogonal unitary matrices
over Grassmannian manifold and the corresponding searching algorithm are investigated.
In this section, we propose the one-way relay system, whose key idea to structure the quasi-EVD channel is using the relay function to cancel the unitary matrices
of multiple access channel (MAC) and broadcast hermitian channel (BHC).
in which [I.sub.t] is a t x t identity matrix, then [P.sub.t], [Q.sub.t], [S.sub.t], [R.sub.t] are unitary matrices
where U = [[u.sub.1], [u.sub.2], ..., [u.sub.M]] and V = [[[upsilon].sub.1], [[upsilon].sub.2], ..., [[upsilon].sub.N]] (having dimensions Q x Q and N x N) are left and right unitary matrices
Let U(N) denote the group of all N x N unitary matrices
. The infinite dimensional unitary group is defined as the inductive limit of U(N)s, where U(N) embeds in U(N +1) by fixing the N + 1st vector:
where V and U are unitary matrices
, that is, [V.sup.-1] = [V.sup.H] and [U.sup.-1] = [U.sup.H] (with H the Hermitian operator) and D is a diagonal matrix containing the singular values of H.