Generators of matrix algebras in dimension 2 and 3

Let K be an algebraically closed field of characteristic zero and consider a set of 2×2 or 3×3 matrices. Using a theorem of Shemesh, we give conditions for when the matrices in the set generate the full matrix algebra.     

Unitary equivalence to a complex symmetric matrix: An algorithm

We present a necessary and sufficient condition for a 3×3 matrix to be unitarily equivalent to a symmetric matrix with complex entries, and an algorithm whereby an arbitrary 3×3 matrix can be tested. This test generalizes to a necessary and sufficient condition that applies to almost every n×n matrix. The test is constructive in that it explicitly exhibits the unitary equivalence to a complex symmetric matrix.     

On the Drazin inverse of block matrices and generalized Schur complement

In this paper, we give an additive result for the Drazin inverse with its applications, we obtain representations for the Drazin inverse of a 2 × 2 complex block matrix having generalized Schur complement S=D-CA^DB equal to zero or nonsingular. Several situations are analyzed and recent results are generalized [R.E. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2×2 block matrix, SIAM J. Matrix Anal. Appl. 27 (3) (2006) 757-771].     

A structure of the Bang-Bang representation for 3×3 embeddable matrices

Summary Necessary and sufficient conditions are given for a 3×3 stochastic matrix to be embeddable by 6 elementary stochastic matrices (Poisson matrices). For a 3×3 embeddable matrix, a structure of the minimal Bang-Bang representation, i.e. the one that contains the smallest number of elementary matrices, is obtained. Based on the minimal Bang-Bang representation an algorithm for determining the embeddability of a 3×3 stochastic matrix is given.I would like to thank Søren Johansen for helpful comments and stimulating discussions on the subject of this paper     

超AKNS方程新的可积分解

该文介绍从3×3矩阵形式超谱问题出发, 构造新高阶矩阵形式超谱问题的方法.以超AKNS方程为例, 作者构造了5×5矩阵形式的超AKNS谱问题并且运用双非线性化方法,给出了超AKNS方程的新约束, 得到该约束下超AKNS方程新的可积分解.     

On the completion of a partial integral matrix to a unimodular matrix

We prove that if a partial integral matrix has a free diagonal then this matrix can be completed to a unimodular matrix. Such a condition is necessary in a general sense. Consequently if an n × n (n ? 2) partial integral matrix has 2n − 3 prescribed entries and any n entries of these do not constitute a row or a column then it can be completed to a unimodular matrix. This improves a recent result of Zhan.     

Half Dirichlet problem for matrix functions on the unit ball in Hermitian Clifford analysis

The simultaneous null solutions of the two complex Hermitian Dirac operators are focused on in Hermitian Clifford analysis, where the Hermitian Cauchy integral was constructed and will play an important role in the framework of circulant (2×2) matrix functions. Under this setting we will present the half Dirichlet problem for circulant (2×2) matrix functions on the unit ball of even dimensional Euclidean space. We will give the unique solution to it merely by using the Hermitian Cauchy transformation, get the solution to the Dirichlet problem on the unit ball for circulant (2×2) matrix functions and the solution to the classical Dirichlet problem as the special case, derive a decomposition of the Poisson kernel for matrix Laplace operator, and further obtain the decomposition theorems of solution space to the Dirichlet problem for circulant (2×2) matrix functions.     

Integrable coupling hierarchy and Hamiltonian structure for a matrix spectral problem with arbitrary-order

We presented an integrable coupling hierarchy of a matrix spectral problem with arbitrary order zero matrix r by using semi-direct sums of matrix Lie algebra. The Hamiltonian structure of the resulting integrable couplings hierarchy is established by means of the component trace identities. As an example, when r is 2 × 2 zero matrix specially, the integrable coupling hierarchy and its Hamiltonian structure of the matrix spectral problem are computed.     

Factorization of singular integer matrices

It is well known that a singular integer matrix can be factorized into a product of integer idempotent matrices. In this paper, we prove that every n  × n (n > 2) singular integer matrix can be written as a product of 3n + 1 integer idempotent matrices. This theorem has some application in the field of synthesizing VLSI arrays and systolic arrays.     

Samuel multiplicity and the structure of semi-Fredholm operators

Two numerical invariants refining the Fredholm index are introduced for any semi-Fredholm operator in such a way that their difference calculates the Fredholm index. These two invariants are inspired by Samuel multiplicity in commutative algebra, and can be regarded as the stabilized dimension of the kernel and cokernel. A geometric interpretation of these invariants leads naturally to a 4×4 uptriangular matrix model for any semi-Fredholm operator on a separable Hilbert space. This model can be regarded as a refined, local version of the Apostol's 3×3 triangular representation for arbitrary operators. Some classical results, such as Gohberg's punctured neighborhood theorem, can be read off directly from our matrix model. Banach space operators are also considered.     

Stochastic independent modal-space control of distributed-parameter systems

A method is presented for solving time-varying independent modal-space Kalman filter equations in terms of 2×2 transition matrices, rather than in terms of the more commonly used 4×4 transition matrix solution technique. The basic method consists of replacing the well-known product form solution for the differential matrix Riccati equation with an alternate solution form which consists of a steady-state plus transient term.     

Linear spaces of matrices of constant rank and instanton bundles

We present a new method to study 4-dimensional linear spaces of skew-symmetric matrices of constant co-rank 2, based on rank 2 vector bundles on P³${\mathbb{P}}^3$ and derived category tools. The method allows one to prove the existence of new examples of size 10×10$10\times 10$ and 14×14$14\times 14$ via instanton bundles of charge 2 and 4 respectively, and it provides an explanation for what used to be the only known example (Westwick 1996 [25]). We also give an algorithm to construct explicitly a matrix of size 14 of this type.     

Minimal and locally minimal games and game forms

By Shapley’s (1964) theorem, a matrix game has a saddle point whenever each of its 2×2 subgames has one. In other words, all minimal saddle point free (SP-free) matrices are of size 2×2. We strengthen this result and show that all locally minimal SP-free matrices also are of size 2×2. In other words, if A is a SP-free matrix in which a saddle point appears after deleting an arbitrary row or column then A is of size 2×2. Furthermore, we generalize this result and characterize the locally minimal Nash equilibrium free (NE-free) bimatrix games.Let us recall that a two-person game form is Nash-solvable if and only if it is tight [V. Gurvich, Solution of positional games in pure strategies, USSR Comput. Math. and Math. Phys. 15 (2) (1975) 74-87]. We show that all (locally) minimal non-tight game forms are of size 2×2. In contrast, it seems difficult to characterize the locally minimal tight game forms (while all minimal ones are just trivial); we only obtain some necessary and some sufficient conditions. We also recall an example from cooperative game theory: a maximal stable effectivity function that is not self-dual and not convex.     

A fully stable rational version of theQR algorithm for tridiagonal matrices

Summary A rational version of theQR algorithm for symmetric tridiagonal matrices is presented. Stability is ensured by calculating the elements of the transformed matrix by various formulas, depending on the angle of rotation. Virtual origin shifts are determined from perturbation estimates for the leading 2×2 and 3×3 submatrices; the size of these shifts can optionally serve as a convergence criterion. A number of test matrices, including one with several degeneracies, were diagonalized; an average of 1.3–1.5QR iterations per eigenvalue was needed for 12-figure precision, and of 1.7–2.0 for 22-figure precision.     

Lax Pairs for the Deformed Kowalevski and Goryachev–Chaplygin Tops

We consider a quadratic deformation of the Kowalevski top. This deformation includes a new integrable case for the Kirchhoff equations recently found by one of the authors as a degeneration. A 5×5 matrix Lax pair for the deformed Kowalevski top is proposed. We also find similar deformations of the two-field Kowalevski gyrostat and the so(p,q) Kowalevski top. All our Lax pairs are deformations of the corresponding Lax representations found by Reyman and Semenov-Tian-Shansky. A similar deformation of the Goryachev–Chaplygin top and its 3×3 matrix Lax representation is also constructed.     

Essential spectra of some matrix operators and application to two-group transport operators with general boundary conditions

In this article we investigate the essential spectra of a 2×2 block operator matrix on a Banach space. Furthermore, we apply the obtained results to determine the essential spectra of two-group transport operators with general boundary conditions in the Banach space Lp([−a,a]×[−1,1])×Lp([−a,a]×[−1,1]), a>0.     

Some properties of generalized K-centrosymmetric H-matrices

Every n×n$n\times n$ generalized K-centrosymmetric matrix A   can be reduced into a 2×2$2\times 2$ block diagonal matrix (see [Z. Liu, H. Cao, H. Chen, A note on computing matrix–vector products with generalized centrosymmetric (centrohermitian) matrices, Appl. Math. Comput. 169 (2) (2005) 1332–1345]). This block diagonal matrix is called the reduced form of the matrix A. In this paper we further investigate some properties of the reduced form of these matrices and discuss the square roots of these matrices. Finally exploiting these properties, the development of structure-preserving algorithms for certain computations for generalized K-centrosymmetric H-matrices is discussed.     

On numerical ranges of the compressions of normal matrices

For an n × n normal matrix A, whose numerical range NR[A] is a k-polygon (k ? n), an n × (k − 1) isometry matrix P is constructed by a unit vector υ∈Cⁿ, and NR[P∗AP] is inscribed to NR[A]. In this paper, using the notations of NR[P∗AP] and some properties from projective geometry, an n × n diagonal matrix B and an n × (k − 2) isometry matrix Q are proposed such that NR[P∗AP] and NR[Q∗BQ] have as common support lines the edges of the k-polygon and share the same boundary points with the polygon. It is proved that the boundary of NR[P∗AP] is a differentiable curve and the boundary of the numerical range of a 3 × 3 matrix P∗AP is an ellipse, when the polygon is a quadrilateral.     

Testing balancedness and perfection of linear matrices

A (0, 1) matrix is linear if it does not contain a 2×2 submatrix of all ones. In this paper we give polynomial algorithms to test whether a linear matrix is balanced or perfect. The algorithms are based on decomposition results previously obtained by the authors.Partial support under NSF Grants DMS 8606188, DDM8800281 and DDM9001705.     

Generalizing the column-row matrix decomposition to multi-way arrays

In this paper, we provide two generalizations of the CUR matrix decomposition Y=CUR (also known as pseudo-skeleton approximation method [1]) to the case of N-way arrays (tensors). These generalizations, which we called Fiber Sampling Tensor Decomposition types 1 and 2 (FSTD1 and FSTD2), provide explicit formulas for the parameters of a rank-(R,R,…,R) Tucker representation (the core tensor of size R×R×?×R and the matrix factors of sizes I_n×R, n=1,2,…N) based only on some selected entries of the original tensor. FSTD1 uses P^N-1(P?R)n-mode fibers of the original tensor while FSTD2 uses exactly R fibers in each mode as matrix factors, as suggested by the existence theorem provided in Oseledets et al. (2008) [2], with a core tensor defined in terms of the entries of a subtensor of size R×R×?×R. For N=2 our results are reduced to the already known CUR matrix decomposition where the core matrix is defined as the inverse of the intersection submatrix, i.e. U=W^-1. Additionally, we provide an adaptive type algorithm for the selection of proper fibers in the FSTD1 model which is useful for large scale applications. Several numerical results are presented showing the performance of our FSTD1 Adaptive Algorithm compared to two recently proposed approximation methods for 3-way tensors.