Sarkar, "2-D

unitary matrix pencil method for efficient direction of arrival estimation," Digital Signal Process., vol.

These techniques are firstly exploited by the unitary MUSIC (U-MUSIC) algorithm [19] and then extended to many other derivations including unitary root MUSIC (U-root-MUSIC) [20], unitary ESPRIT (U-ESPRIT) [21], unitary method of direction-of-arrival estimation (U-MODE) [22], and

unitary matrix pencil (U-MP) [23].

To circumvent deflation at startup, the subspace decomposition at the beginning of the cycle is derived by finding a

unitary matrix [F.sub.1] [member of] [R.sup.sxs], such that

For convenience, choose T = [[[U.sub.2] [U.sub.1]].sup.T], where [U.sub.1] [member of] [R.sup.nxm], [U.sub.2] [member of] [R.sup.nx(n-m)] are two subblocks of a

unitary matrix resulting from the singular value decomposition of [??], i.e.,

Thereafter, we can obtain the rectangular diagonal matrix S and

unitary matrix U, V.

where [U.sub.k] is the principal submatrix of order k of U obtained from the Krylov matrix K and the

unitary matrix V through KU = V, and [C.sub.(k)] is the companion matrix for the Ritz values at iteration k.

By the well known Schur's triangularization theorem, for any n by n complex matrix A in M(n), there is a

unitary matrix U such that U*AU (also in M(n)) is upper triangular.

Since the capacity-achieving USTM signal distribution at high SNR is isotropic on the Grassmannian manifold [G.sub.T,M] and each signal point is denoted as a

unitary matrix [PHI], [bar.[PHI]] is defined as an antipodal point of [PHI] if [bar.[PHI]] is the orthogonal complement of [PHI] on [G.sub.T,M].

where the vector eig(M) contains the eigenvalues of the square matrix M, WA is an

unitary matrix, and [[theta].sub.i], for i = 1, ..., n are the eigenvalues of [R.sup.[xi].sub.s]([tau]).

where U is an m * m

unitary matrix over K, [summation] is a m x n diagonal matrix with non-negative real numbers on the diagonal.

where L is a lower triangular matrix and Q is a

unitary matrix. The feedback, feed forward and scaling filters for the THP algorithm can be obtained as,

Similarly, some vectors were omitted from the model because of poor quality or methodological disturbances, while some were added with the goal of improving the forecast--the weather vector was multiplied by a

unitary matrix creating 4 new vectors representing different consumption behavior in different parts of the year: