Unitary Triangularization Research Articles

Joint Unitary Triangularization for Gaussian Multi-User MIMO Networks

The problem of transmitting a common message to multiple users over the Gaussian multiple-input multiple-output broadcast channel is considered, where each user is equipped with an arbitrary number of antennas. A closed-loop scenario is assumed, for which a practical capacity-approaching scheme is developed. By applying judiciously chosen unitary operations at the transmit and receive nodes, the channel matrices are triangularized so that the resulting matrices have equal diagonals, up to a possible multiplicative scalar factor. This, along with the utilization of successive interference cancellation, reduces the coding and decoding tasks to those of coding and decoding over the single-antenna additive white Gaussian noise channel. Over the resulting effective channel, any off-the-shelf code may be used. For the two-user case, it was recently shown that such joint unitary triangularization is always possible. In this paper, it is shown that for more than two users, it is necessary to carry out the unitary linear processing jointly over multiple channel uses, i.e., space-time processing is employed. It is further shown that exact triangularization, where all resulting diagonals are equal, is still not always possible, and appropriate conditions for the existence of such are established for certain cases. When exact triangularization is not possible, an asymptotic construction is proposed, that achieves the desired property of equal diagonals up to edge effects that can be made arbitrarily small, at the price of processing a sufficiently large number of channel uses together.

Riesz operators and Schur’s lemma

We present sucient conditions in order (the space of) a Riesz op- erator T in a Hilbert space H have a Jordan-Schur basis with respect to a scalar product equivalent to the original one. This is related to Schur's lemma for a compact operator, which is an extension of Schur's classical theorem on unitary triangularization in a finite dimensional space. The finite dimensional case is also studied.

Cluster synchronization in a network of non-identical dynamic systems

Cluster synchronization in a network of non-identical dynamic systems is studied in this paper, using two-cluster synchronization for detailed analysis and discussion. The results show that the common intercluster coupling condition is not always needed for the diffusively coupled network. Several sufficient conditions are obtained by using the Schur unitary triangularization theorem, which extends previous results. Some numerical examples are presented for illustration.

Joint Unitary Triangularization for MIMO Networks

This work considers communication networks where individual links can be described as MIMO channels. Unlike orthogonal modulation methods (such as the singular-value decomposition), we allow interference between sub-channels, which can be removed by the receivers via successive cancellation. The degrees of freedom earned by this relaxation are used for obtaining a basis which is simultaneously good for more than one link. Specifically, we derive necessary and sufficient conditions for shaping the ratio vector of sub-channel gains of two broadcast-channel receivers. We then apply this to two scenarios: First, in digital multicasting we present a practical capacity-achieving scheme which only uses scalar codes and linear processing. Then, we consider the joint source-channel problem of transmitting a Gaussian source over a two-user MIMO channel, where we show the existence of non-trivial cases, where the optimal distortion pair (which for high signal-to-noise ratios equals the optimal point-to-point distortions of the individual users) may be achieved by employing a hybrid digital-analog scheme over the induced equivalent channel. These scenarios demonstrate the advantage of choosing a modulation basis based upon multiple links in the network, thus we coin the approach "network modulation".

On similarities of class Cp and applications to matrix differential equations

This paper deals with similarities of class C p , in particular reduction of class C p to Jordan form and unitary triangularization of class C p of a matrix function. The main result is applied to establish a method to lower the order of nonlinear matrix differential equations belonging to a broad class of equations. In particular, this method permits one to transform nonlinear equations of the first order into an algebraic equation joint to a linear differential equation of the first order. The method is applied to the matrix differential equation XX′ − X′ X = X, and a particular solution is easily obtained.

Unit�re Transformationen gro�er Matrizen

Strassen [2] has described a method for the multiplication of (N, N)-matrices which needs O (N 2.8...) basic operations. Here algorithms are given forQR-decomposition and unitary transformation of arbitrary complex matrices to upper Hessenberg form and for unitary triangularization of hermitean matrices, which by use of a fast matrix multiplication with time bound O (N ?) have nearly the same speed.

On the Continuity of the Generalized Inverse

On the Continuity of the Generalized Inverse

Remarks on the Unitary Triangularization of a Nonsymmetric Matrix

In [1], A. Householder described a method for the unitary triangularization of a matrix. The formulas given there are valid for the real case. In this note we describe the modifications to handle the complex case and also point out a small modification in the real case which will improve the numerical accuracy of the method. At first we are concerned with a complex vector space. The basic tool is the fact that if ‖ u ‖ = √2, then the matrix I - uu * is unitary, as may be readily verified. The following lemma is a modification of the one give in [1]. LEMMA. Let a ≠ 0 be an arbitrary vector and let v be an arbitrary unit vector. Then there exists a vector u with ‖ u ‖ = √2 and a scalar ζ with | ζ | = 1 such that ( I - uu * ) a = ζ ‖ a ‖ v . (1) PROOF: Letting α = ‖ a ‖ and μ = u * a , (1) may be written a - αζv = μu . (2) Multiply by a * gives α 2 - αζa * v = μa * u = ‖ μ ‖ 2 . (3) It follows that ζa * v is real. Assuming for the moment that a * v ≠ 0, we write it in polar form a * v = rw , r > 0, ‖ w ‖ = 1. Then the fact that ζrw is real implies that ζ = ± w . (4) Substituting into (3) gives ‖ μ ‖ 2 = α 2 ∓ αr . (5) We now set, arbitrarily, arg ( μ ) = 0. Then μ = √ α ( α ∓ r ). (6) Next, we select the negative sign in (4) in order to avoid the subtraction of two positive quantities in (6), since such a subtraction may give rise to numerical difficulties. Collecting the formulas, we see that the following sequence of computations will produce the required u and ζ : α = ‖ a ‖ (7) r = ‖ a * v ‖ (8) ζ = - a * v / r (9) μ = √ α ( α + r ) (10) u = 1/ μ ( a - ζαv ). (11) The case a * v = 0 may be handled if, instead of using (9), we let ζ be an arbitrary number with ‖ ζ ‖ = 1. It is easily verified that the formulas thus modified will still work. The computation requires 3 square roots to compare α , r , and μ . A slight modification [1] permits one to avoid the root required to compute μ . In the real case, no root is required to compute r . Now consider the case of a real vector space. The formulas given [1] for this case are essentially the same as ours except that (8) is replaced by r = a * v , and (10) by μ = √ α ( α - r ). If μ is computed this way and if r is positive and near α (as is the case when v is near a / α ), cancellation of significant digits will occur. This difficulty, and the need for making a special case when v is exactly a / α , is avoided in the present set of formulas.

Unitary Triangularization of a Nonsymmetric Matrix

A method for the inversion of a nonsymmetric matrix has been in use at ORNL and has proved to be highly stable numerically but to require a rather large number of arithmetic operations, including a total of N(N-1)/2 square roots. This note points out that the same result can be obtained with fewer arithmetic operations, and, in particular, for inverting a square matrix of order N, at most 2(N-1) square roots are required. For N > 4, this is a savings of (N-4)(N-1)/4 square roots. (T.B.A.)