Distinct Singular Values Research Articles

Bi-Hamiltonian structure of Sutherland models coupled to two -valued spins from Poisson reduction

We introduce a bi-Hamiltonian hierarchy on the cotangent bundle of the real Lie group , and study its Poisson reduction with respect to the action of the product group U(n) × U(n) arising from left- and right-multiplications. One of the pertinent Poisson structures is the canonical one, while the other is suitably transferred from the real Heisenberg double of . When taking the quotient of we focus on the dense open subset of whose elements have pairwise distinct singular values. We develop a convenient description of the Poisson algebras of the U(n) × U(n) invariant functions, and show that one of the Hamiltonians of the reduced bi-Hamiltonian hierarchy yields a hyperbolic Sutherland model coupled to two -valued spins. Thus we obtain a new bi-Hamiltonian interpretation of this model, which represents a special case of Sutherland models coupled to two spins obtained earlier from reductions of cotangent bundles of reductive Lie groups equipped with their canonical Poisson structure. Upon setting one of the spins to zero, we recover the bi-Hamiltonian structure of the standard hyperbolic spin Sutherland model that was derived recently by a different method.

Graphs with few distinct eigenvalues and extremal energy

In his survey “Beyond graph energy: Norms of graphs and matrices” [23], Nikiforov proposed two problems concerning characterizing the graphs that attain equality in a lower bound and in a upper bound for the energy of a graph, respectively. We show that these graphs have at most two nonzero distinct singular values and investigate the proposed problems organizing our study according to the type of spectrum they can have. In most cases all graphs are characterized. Infinite families of graphs are given otherwise. We also show that all graphs satisfying the properties required in the problems are integral, except for complete bipartite graphs Kp,q and disconnected graphs with a component Kp,q, where pq is not a perfect square.

On the dense subsets of matrices with distinct eigenvalues and distinct singular values

It is well known that the set of all $n \times n$ matrices with distinct eigenvalues is a dense subset of the set of all real or complex $n \times n$ matrices. In [D.J. Hartfiel. Dense sets of diagonalizable matrices. {\em Proceedings of the American Mathematical Society}, 123(6):1669--1672, 1995.], the author established a necessary and sufficient condition for a subspace of the set of all $n \times n$ matrices to have a dense subset of matrices with distinct eigenvalues. The objective of this article is to identify necessary and sufficient conditions for a subset of the set of all $n \times n$ real or complex matrices to have a dense subset of matrices with distinct eigenvalues. Some results of Hartfiel are extended, and the existence of dense subsets of matrices with distinct singular values in the subsets of the set of all real or complex matrices is studied. Furthermore, for a matrix function $F(x)$, defined on a closed and bounded interval whose entries are analytic functions, it is proved that the set of all points for which the matrix $F(x)$ has repeated analytic eigenvalues/analytic singular values is either a finite set or the whole domain of the function $F$.

Refined bounds on the number of distinct eigenvalues of a matrix after low-rank update

ABSTRACT The eigenproblems of low-rank updated matrices are of crucial importance in many applications. Recently, an upper bound on the number of distinct eigenvalues of a low-rank updated matrix was established in [Farrell P. SIAM J Matrix Anal Appl. 2016:37;572–576]. The result can be applied to estimate the number of Krylov iterations required for solving a perturbed linear system. In this paper, we revisit this problem and establish some refined bounds. Some a prior upper bounds that only rely on the information of the matrix in question and the low-rank update are provided. Examples show the superiority of our theoretical results over the existing ones. The number of distinct singular values of a matrix after perturbation is also investigated.

Convexity and star-shapedness of real linear images of special orthogonal orbits

Let A∈Rn×n and SOn:={U∈Rn×n:UUt=In,detU>0} be the set of n×n special orthogonal matrices. Define the (real) special orthogonal orbit of A byO(A):={UAV:U,V∈SOn}. In this paper, we show that the linear image of O(A) is star-shaped with respect to the origin for arbitrary linear maps L:Rn×n→Rℓ if n≥2ℓ−1. In particular, for linear maps L:Rn×n→R2 and when A has distinct singular values, we study B∈O(A) such that L(B) is a boundary point of L(O(A)). This gives an alternative proof of a result by Li and Tam on the convexity of L(O(A)) for linear maps L:Rn×n→R2.

Optimal multi‐input–multi‐output relay transform matrix with a bounded condition number

A relay communication system where its nodes are equipped with multiple antennas is considered. The relay applies a transformation matrix on its received signal and forwards it to the destination. The authors show that, given a limited relay transmit power, the transformation matrix which maximises the signal-to-noise power ratio (SNR) is almost surely a rank-one matrix. Such a singular transformation matrix prevents accurate channel estimation. As a remedy, they impose a constraint on the condition number (CN) of the relay transformation matrix. They prove that the optimal relay matrix under this new constraint has up to two distinct singular values. Thus, the relay transmits some power in all directions, and this small energy allows better channel estimation. The author's simulation results show that imposing the CN constraint on the relay matrix results in some small loss in the SNR which is quantified in this study; however, the optimal relay matrix enables the receivers to track the channel state information.

Continuous decomposition of quantum measurements via qubit-probe feedback

It is known that any two-outcome quantum measurement can be decomposed into a continuous stochastic process using a feedback loop. In this article, we characterize which of these decompositions are possible when each iteration of the feedback loop consists of a weak measurement caused by an interaction with a probe system. We restrict ourselves to the case when the probe is a qubit and the interaction Hamiltonian between the probe and system is constant. We find that even given the ability to perform arbitrary unitary pulses throughout the continuous decomposition, only generalized measurements with two distinct singular values are achievable. However, this is sufficient to decompose a generalized qubit measurement using a qubit probe and a simple interaction Hamiltonian.

Singular value decomposition for the Takagi factorization of symmetric matrices

We describe a simple implementation of the Takagi factorization of symmetric matrices A=UΛUT with unitary U and diagonal Λ≥0 in terms of the square root of an auxiliary unitary matrix and the singular value decomposition of A. The method is based on an algebraically exact expression. For parameterized family Aε=A+εR=UεΛεUεT, ε≥0 with distinct singular values, the unitary matrices Uε are discontinuous at the point ε=0, if the singular values of A are multiple, but the composition UεΛεUεT remains numerically stable and converges to A. The factorization is represented as a fast and compact algorithm. Its demo version for Wolfram Mathematica and interactive numerical tests are available on Internet.

Combinatorial designs with two singular values II. Partial geometric designs

In this and an earlier paper [E.R. van Dam, E. Spence, Combinatorial designs with two singular values. I. Uniform multiplicative designs, J. Combin. Theory A 107 (2004) 127–142] we study combinatorial designs whose incidence matrix has two distinct singular values. These generalize ( v, k, λ) designs, and include uniform multiplicative designs and partial geometric designs. Here we study the latter, which are precisely the designs with constant replication and block size. We collect most known results, give new characterization results, and we enumerate, partly by computer, all small ones.

Combinatorial Designs with Two Singular Values II. Partial Geometric Designs

In this and an earlier paper [17], we study combinatorial designs whose incidence matrix has two distinct singular values. These generalize (v, k, lambda) designs, and include uniform multiplicative designs and partial geometric designs. Here, we study the latter, which are precisely the designs with constant replication and block size. We collect most known results, give new characterization results, and we enumerate, partly by computer, all small ones.

Combinatorial designs with two singular values—I: uniform multiplicative designs

In this and a sequel paper (Combinatorial designs with two singular values. II. Partial geometric designs, preprint) we study combinatorial designs whose incidence matrix has two distinct singular values. These generalize 2-( v, k, λ) designs, and include partial geometric designs and uniform multiplicative designs. Here we study the latter, which are precisely the nonsingular designs. We classify all such designs with smallest singular value at most 2 , generalize the Bruck–Ryser–Chowla conditions, and enumerate, partly by computer, all uniform multiplicative designs on at most 30 points.

Combinatorial designs with two singular values?I: uniform multiplicative designs

In this and a sequel paper (Combinatorial designs with two singular values. II. Partial geometric designs, preprint) we study combinatorial designs whose incidence matrix has two distinct singular values. These generalize 2-( v , k , λ ) designs, and include partial geometric designs and uniform multiplicative designs. Here we study the latter, which are precisely the nonsingular designs. We classify all such designs with smallest singular value at most 2 , generalize the Bruck–Ryser–Chowla conditions, and enumerate, partly by computer, all uniform multiplicative designs on at most 30 points.

On variable blocking factor in a parallel dynamic block––Jacobi SVD algorithm

The parallel two-sided block-Jacobi singular value decomposition (SVD) algorithm with dynamic ordering, originally proposed in [Parallel Comput. 28 (2002) 243–262], has been extended with respect to the blocking factor ℓ. Unlike the unique blocking factor ℓ=2 p in the original algorithm running on p processors, the current blocking factor is a variable parameter that covers the values in two different regions––namely, ℓ= p/ k and ℓ=2 kp for some integer k. Two new parallel two-sided block-Jacobi SVD algorithms with dynamic ordering are described in detail. They arise in those two regions and differ in the logical data arrangement and communication complexity of the reordering step. For the case of ℓ=2 kp, it is proved that a designed point-to-point communication algorithm is optimal with respect to the amount of communication required per processor as well as to the amount of overall communication. Using the message passing programming model for distributed memory machines, new parallel block-Jacobi SVD algorithms were implemented on an SGI-Cray Origin 2000 parallel computer. Numerical experiments were performed on p=12 and 24 processors using a set of six matrices of order 4000 and blocking factors ℓ, 2⩽ℓ⩽192. To achieve the minimal total parallel execution time, the use of a blocking factor ℓ∈{2, p,2 p} can be recommended for matrices with distinct singular values. However, for matrices with a multiple minimal singular value, the total parallel execution time may monotonically increase with ℓ. In this case, the recommended Jacobi method with ℓ=2 is just the ScaLAPACK routine with some additional matrix multiplications, and it computes the SVD in one parallel iteration step.

Combinatorial Designs with Two Singular Values I. Uniform Multiplicative Designs

In this and a sequel paper, we study combinatorial designs whose incidence matrix has two distinct singular values. These generalize 2-(v, k, lambda) designs, and include partial geometric designs and uniform multiplicative designs. Here, we study the latter, which are precisely the nonsingular designs. We classify all such designs with smallest singular value at most square root 2, generalize the Bruck-Ryser-Chowla conditions, and enumerate, partly by computer, all uniform multiplicative designs on at most 30 points.

A parameterization of minimal plants

We present the input-normal Riccati parameterization, a generalization of Kabamba's input-normal form, which allows the continuous parameterization of the set of minimal linear systems of a given order that have distinct singular values; the input-normal Riccati form has no requirement that the underlying system be stable. We also present formulas for the use of the input-normal Riccati parameterization for the synthesis of systems via gradient methods. >

Numerical computation of an analytic singular value decomposition of a matrix valued function

This paper extends the singular value decomposition to a path of matricesE(t). An analytic singular value decomposition of a path of matricesE(t) is an analytic path of factorizationsE(t)=X(t)S(t)Y(t) T whereX(t) andY(t) are orthogonal andS(t) is diagonal. To maintain differentiability the diagonal entries ofS(t) are allowed to be either positive or negative and to appear in any order. This paper investigates existence and uniqueness of analytic SVD's and develops an algorithm for computing them. We show that a real analytic pathE(t) always admits a real analytic SVD, a full-rank, smooth pathE(t) with distinct singular values admits a smooth SVD. We derive a differential equation for the left factor, develop Euler-like and extrapolated Euler-like numerical methods for approximating an analytic SVD and prove that the Euler-like method converges.

Phase and amplitude calibration in aperture synthesis. Algebraic structures

The inverse problems of image reconstruction in aperture synthesis are very particular myopic deconvolution problems. The constraints on the transfer function, the calibration constraints, can easily be imposed. This results from the particular algebraic structures of interferometry. The real and imaginary parts of the logarithm of the transfer function lie in the ranges of two remarkable operators: the amplitude aberration operator Brho and the phase aberration operator Bphi . These operators have only two distinct singular values: square root 2n-2, square root n-2 for the first, and 0, square root n for the second (n is the number of antennas or pupil elements of the interferometric device). As a corollary, the projections onto the ranges of Brho and Bphi can be expressed in very simple closed forms. These very nice properties, which had so far remained unknown, reveal in fact more general structures underlying phase and amplitude compilation (and alternate compilation).

Balanced forms: Canonicity and parameterization

We show that balanced forms are canonical and continuously parameterize the set of systems of a given order which are asymptotically stable, minimal, and have distinct singular values. The advantage of the balanced canonical form over other classical canonical forms lies precisely in the guaranteed stability and minimality of the parameterization. We also derive formulas allowing the synthesis of systems using gradient methods in balanced form.

A Lanczos Algorithm for Computing Singular Values and Vectors of Large Matrices

Any real matrix A has associated with it the real symmetric matrix \[ B \equiv \left(\begin{array}{*{20}c} 0 \\ {A^T } \\ \end{array} \begin{array}{*{20}c} A \\ 0 \\ \end{array} \right) \] whose positive eigenvalues are the nonzero singular values of A. Using B and our Lanczos algorithms for computing eigenvalues and eigenvectors of very large real symmetric matrices, we obtain an algorithm for computing singular values and singular vectors of large sparse real matrices. This algorithm provides a means for computing the largest and the smallest or even all of the distinct singular values of many matrices