Inverting linear combinations of identity and generalized Catalan matrices

The expression of a bound of the uniform norm of infinite lower triangular Toeplitz matrices with nonnegative entries is found. All the results are obtained by studying the behavior of the resolvent kernel and of the fundamental matrix of the recurrence relation, which generates the sequence of the entries of the considered matrix.

Catalan matrix and related combinatorial identities

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper L Subscript upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">L</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {L}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the collection of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N times upper N"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo> × </mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">N\times N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> lower triangular Toeplitz matrices and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper T Subscript upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {T}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the collection of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N times upper N"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo> × </mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">N\times N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> lower triangular Toeplitz contractions. We show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper T Subscript upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {T}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is compact and strictly convex, in the spectral norm, with respect to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper L Subscript upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">L</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {L}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; that is, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper T Subscript upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {T}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is compact, convex and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Subscript German upper L Sub Subscript upper N Baseline German upper T Subscript upper N subset-of-or-equal-to e x t German upper T Subscript upper N"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">L</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msub> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:mo> ⊆ </mml:mo> <mml:mi>ext</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial _{\mathfrak {L}_{N}} \mathfrak {T}_{N} \subseteq \operatorname {ext}\mathfrak {T}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Subscript German upper L Sub Subscript upper N Baseline left-parenthesis dot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">L</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msub>

Displacement operator based decompositions of matrices using circulants or other group matrices

The classical Schur extension problem is to describe the set of contractive holomorphic functions s(ζ) in the open unit disk of the form It is well‐known that this set is nonempty if and only if the (n + 1) × (n + 1) lower triangular Toeplitz matrix A with entries aij = ai − j is contractive, i.e., if and only if In + 1 − AA* ≥ 0; see e.g., p. 79 of 5. In this paper the analogue of this problem for mvf's (matrix valued functions) that are holomorphic and contractive in the open upper half plane and bitangential generalizations thereof are studied.

Toeplitz and Laurent operators play a basic role in systems theory. For example, a lower triangular Toeplitz matrix can be viewed as an input-output map for a linear causal time invariant system. First we introduce the Fourier transform. Then we will study Toeplitz and Laurent operators. The Fourier transform will be used to turn Laurent operators into multiplication operators and visa versa.

We use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict k-Hessenberg matrices and banded matrices. Our results can be extended to the cases of block triangular and block Hessenberg matrices. An n × n lower triangular matrix is called elementary if it is of the form I + C, where I is the identity matrix and C is lower triangular and has all of its nonzero entries in the k-th column,where 1 ≤ k ≤ n.

This study develops and tests a MATLAB program for neutralising systems with upper triangular coefficient matrices. Augmented matrices for systems of linear equations are reduced to produce a neutral (identity matrix) left hand side, at which point the right hand side has the solutions for the system. A strategy for minimal error is pursued, where elimination operations must be completed in one step for each number. The strategy was expressed in pseudocode, a flowchart developed, and then the program was coded in MATLAB. The program was tested for four sample augmented matrices of varying sizes. The results demonstrated that the program was 100% accurate. The technique can be applied to any square matrix system, since any square matrix can be expressed as an upper triangular matrix. The automation of matrix operations and achievement of 100% accuracy allows for the use of the program for very large data sets, extending the potential for research with accurate finding.

Updating URV decompositions in parallel

Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This second edition of this acclaimed text presents results of both classic and recent matrix analysis using canonical forms as a unifying theme and demonstrates their importance in a variety of applications. This thoroughly revised and updated second edition is a text for a second course on linear algebra and has more than 1,100 problems and exercises, new sections on the singular value and CS decompositions and the Weyr canonical form, expanded treatments of inverse problems and of block matrices, and much more.

A QR decomposition aided belief propagation (QR‐BP) detector for multiple‐input multiple‐output (MIMO) systems is proposed. Before the iterative message passing process, linear transformation is performed by using the unitary matrix obtained from the QR decomposition of the original MIMO channel matrix. Then, the original MIMO channel matrix is replaced by an upper triangular channel matrix after the linear transformation and the proposed QR‐BP detector operates over the new bipartite graph from the upper triangular channel matrix. Therefore, the number of cycles in the new bipartite graph is significantly decreased when compared with the fully connected bipartite graph from the original MIMO channel matrix. In addition, since the unitary matrix is utilised for the linear transformation, there is no information loss on the likelihoods estimated from noisy observations. Simulation results verify that the proposed QR‐BP detector outperforms existing BP detectors and achieves near‐optimum performance close to the maximum‐likelihood detector in terms of bit error ratio.

This article is a sequel to the earlier articles, which describe the invertible ordinary differential operators and their generalizations. The generalizations are invertible mappings of filtered modules generated by one differentiation, and are called invertible D-operators. In particular, invertible ordinary linear differential operators, invertible linear difference operators with periodic coefficients, maps defined by unimodular matrices, and C-transformations of control systems are invertible D-operators. C-Transformations are those invertible transformations for which the variables of one system are expressed in terms of the variables of the other system and their derivatives.In the article we consider the invertible D-operators whose inverses are D-operators of the same type. In previous papers, a classification of invertible D-operators was obtained. Namely, a table of integers was associated to each invertible D-operator. These tables were described in a clear elementary-geometric language. Thus, to each invertible D-operator one assigns an elementary-geometric model, which is called a d-scheme of squares. The class of invertible D-operators having the same d-scheme was also described. In this paper, the invertible D-operators whose d-schemes consist of a single square are called unicellular. It is proved that any unicellular operator in some bases is given by an upper triangular matrix that differs from the identity matrix only by the first row. The main result is representation of the arbitrary invertible D-operator as a composition of unicellular operators. The minimum number of unicellular operators in such a composition is equal to the number of squares of the d-scheme of the original D-operator. As in previous papers, the used method is based on the description of d-schemes in the language of spectral sequences of algebraic complexes.The results obtained can be useful in the transformation and classification of control systems, in particular to describe flat systems.

We study the densities of limiting distributions of squared singular values of high-dimensional matrix products composed of independent complex Gaussian (complex Ginibre) and truncated unitary matrices which are taken from Haar distributed unitary matrices with appropriate dimensional growth. In the general case, we develop a new approach to obtain complex integral representations for densities of measures whose Stieltjes transforms satisfy algebraic equations of a certain type. In the special cases in which at most one factor of the product is a complex Gaussian, we derive elementary expressions for the limiting densities using suitable parameterizations for the spectral variable. Moreover, in all cases we study the behavior of the densities at the boundary of the spectrum.

13 - Matrices and related results

The Pascal matrix, which is related to Pascal’s triangle, appears in many places in the theory of uniform distribution and in many other areas of mathematics. Examples are the construction of low-discrepancy sequences as well as normal numbers or the binomial transforms of Hankel matrices. Hankel matrices which are defined by Catalan numbers and related to the paperfolding sequence are interesting objects in number theory. Therefore, matrices that share many properties with the Pascal matrix or such Hankel matrices are of interest. In this note we will collect common features of the Pascal matrix and the same modulo 2 as well as the Hankel matrix defined by Catalan numbers once pure and once modulo 2 in the ring of integers. Hankel matrices with only 0 and 1 entries in e.g., finite fields gave recently access to counterexamples to the so-called X-adic Liouville conjecture. This justifies as well as motivates our consideration of further matrices with 0 and 1 entries.