The rank inequality for diagonally magic matrices

A new class of general-base matrices, named sampling matrices, which are meant to bridge the gap between algorithmic description and computer architecture is proposed. Poles, zeros, pointers, and spans are among the terms introduced to characterize properties of this class of matrices. A formalism for the decomposition of a general matrix in terms of general-base sampling matrices is proposed. Span matrices are introduced to measure the dependence of a matrix span on algorithm parameters and, among others, the interaction between this class of matrices and the general-base perfect shuffle permutation matrix previously introduced. A classification of general-base parallel recirculant and parallel pipelined processors based on memory topology, access uniformity and shuffle complexity is proposed. The matrix formalism is then used to guide the search for algorithm factorizations leading to optimal parallel and pipelined processor architecture. >

MDS matrices incorporate diffusion layers in block ciphers and hash functions. MDS matrices are in general not sparse and have a large description and thus induce costly implementations both in hardware and software. It is also nontrivial to find MDS matrices which could be used in lightweight cryptography. In the AES MixColumn operation, a circulant MDS matrix is used which is efficient as its elements are of low hamming weights, but no general constructions and study of MDS matrices from d×d circulant matrices for arbitrary d is available in the literature. In a SAC 2004 paper, Junod et al. constructed a new class of efficient matrices whose submatrices were circulant matrices and they coined the term circulating-like matrices for these new class of matrices. We call these matrices as Type-I circulant-like matrices. In this paper we introduce a new type of circulant-like matrices which are involutory by construction and we call them Type-II circulant-like matrices. We study the MDS properties of d×d circulant, Type-I and Type-II circulant-like matrices and construct new and efficient MDS matrices which are suitable for lightweight cryptography for d up to 8. We also consider orthogonal and involutory properties of such matrices and study the construction of efficient MDS matrices whose inverses are also efficient. We explore some interesting and useful properties of circulant, Type-I and Type-II circulant-like matrices which are prevalent in many parts of mathematics and computer science.

In this article, we analyze the circulant structure of generalized circulant matrices to reduce the search space for finding lightweight MDS matrices. We first show that the implementation of circulant matrices can be serialized and can achieve similar area requirement and clock cycle performance as a serial-based implementation. By proving many new properties and equivalence classes for circulant matrices, we greatly reduce the search space for finding lightweight maximum distance separable MDS circulant matrices. We also generalize the circulant structure and propose a new class of matrices, called cyclic matrices, which preserve the benefits of circulant matrices and, in addition, have the potential of being self-invertible. In this new class of matrices, we obtain not only the MDS matrices with the least XOR gates requirement for dimensions from $$3 \times 3$$ to $$8 \times 8$$ in $${\text {GF}}2^4$$ and $${\text {GF}}2^8$$, but also involutory MDS matrices which was proven to be non-existence in the class of circulant matrices. To the best of our knowledge, the latter matrices are the first of its kind, which have a similar matrix structure as circulant matrices and are involutory and MDS simultaneously. Compared to the existing best known lightweight matrices, our new candidates either outperform or match them in terms of XOR gates required for a hardware implementation. Notably, our work is generic and independent of the metric for lightweight. Hence, our work is applicable for improving the search for efficient circulant matrices under other metrics besides XORi¾?gates.

Consider the linear complementarity problem given in the system: [Formula: see text] where, W, Z and q are vectors of dimension n. M is a matrix of order n × n and ZT is the transpose of Z. Any (Z, W) satisfying (1), (2), and (3) is a complementary feasible solution to system (I). In the literature, a class of matrices is defined such that if M belongs to this class, then existence of a feasible solution to system (I) implies the existence of a complementary feasible solution to system (I) with W = 0. In this paper, a new class of matrices 𝔐 is developed. It is shown that membership of a matrix M in 𝔐 is equivalent to the property; for any q existence of a feasible solution to system (I) implies the existence of complementary feasible solution to system (I) for that q with W = 0. This new class of matrices is not contained in any one of the known classes, namely, copositive plus, positive definite or semidefinite, P-matrices, P-matrices, Z-class, etc.

A new class of matrices and its applications to absolute summability factor theorems

This paper deals with the structural properties of lattice-ladder realization of digital filters in a state-space model context. A salient feature of the lattice-ladder realization in state space is its close connection to the Schwarz form, a class of matrices which plays a key role in the stability analysis of linear systems. This connection is further investigated and generalized in this paper to yield a new class of matrices called a generalized Schwarz form. Based on the recursive structure of this new class of matrices, it is shown that there are <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^{n}- 1</tex> lattice realizations of a given digital filter of order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> , each of which corresponds to a different way of connecting lattice sections. Some interesting algebraic properties of generalized lattice realizations are derived which recast the input/output properties of lattice-ladder form from the state-space point of view. Practical advantages of using the generalized lattice form are discussed and a design example is illustrated.

We consider a new class of matrices associated with a real square matrix [Formula: see text] and a vector [Formula: see text] such that [Formula: see text] by using a map [Formula: see text] which turns out to be a conjugation of a matrix [Formula: see text] by a signature matrix. It is shown that every such matrix is similar and congruent to a matrix [Formula: see text] and that they have the same permanental polynomials. There are [Formula: see text] maps [Formula: see text] and they form an abelian group under the composition of maps isomorphic to the group [Formula: see text]. A decomposition of matrices into a sum of symmetric and antisymmetric part under a map [Formula: see text] is considered. Particularly, it is shown that the sum of all principal minors of order [Formula: see text] of a matrix [Formula: see text] is equal to the sum of all principal minors of order [Formula: see text] of their symmetric and antisymmetric parts. It is shown that any symmetric matrix and any antisymmetric matrix under the map [Formula: see text] are simultaneously permutation similar to certain block matrices whose have [Formula: see text] blocks. Finally, for a fixed matrix [Formula: see text], it is proved that the number of different matrices [Formula: see text] is [Formula: see text], where [Formula: see text] is the number of connected components of the graph [Formula: see text] whose adjacency matrix is [Formula: see text].

On a class of matrices with low displacement rank

A study is made of a class of real symmetric matrices, and a new set of necessary and sufficient conditions are defined on the entries of these matrices such that they can be synthesized, without using ideal transformers, as an R-network with a minimum number of terminals. The conditions are stated in terms of the terminal graph used in representing the terminal characteristics of multiterminal components. The new class of matrices then represents "terminal equations" corresponding to a path-tree terminal graph. It is also shown that one of the well-known class of matrices, referred to as dominant matrices, represent the terminal equations for a Lagrangian-tree terminal graph. It is further indicated that any such class of real symmetric matrices is distinguishable by a particular terminal graph. For the cases when a real symmetric matrix of order n cannot be synthesized by an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> network with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n + 1)</tex> terminal vertices, an "enlarged" matrix is formed, and the necessary and sufficient conditions for realizability of these matrices are given.

A very general class of balanced matrices is defined in this paper. These matrices are generalisations of the balanced orthogonal designs of Rao (1970) and generalised balanced matrices of Dey and Midha (1976). Some methods of construction of these balanced matrices are discussed. An application of these matrices in experimental designs is also given.

The characterization of Q-matrices, within the class of P0 matrices due to Aganagic and Cottle, is well known. Afterward, Pang proved a similar characterization for the class L which does not contain class P0. In this note, we establish furher the same result of Pang for a new class of matrices which properly contains class L. Furthermore, the equivalence between a Q-matrix and a Qb-matrix, which consists of matrices such that the linear complementarity problem LCP(q,M) has a nonempty and compact solution set for all \(q\in \mathbb{R}^{n}\), is discussed within such a new class. Positive subdefinite matrices with rank one are specially analyzed.

A new class of matrices with positive inverses

Stair matrices and their generalizations are introduced. The definitions and some properties of the matrices were first given by Lu Hao. this class of matrices provide bases of matrix splittings for iterative methods. The remarkable feature of iterative methods based on the new class of matrices is that the methods are easily implemented for parallel computation. In particular, a generalization of the accelerated overrelaxation method (GAOR) is introduced. Some theories of the AOR method are extended to the generalized method to include a wide class of matrices. The convergence of the new method is derived for Hermitian positive definite matrices. Finally, some examples are given in order to show the superiority of the new method.

Stair matrices and their generalizations are introduced. Some properties of the matrices are presented. Like triangular matrices this class of matrices provides bases of matrix splittings for iterative methods. A remarkable feature of iterative methods based on the new class of matrices is that the methods are easily implemented for parallel computation. In particular, a generalization of the successive overrelaxation (SOR) method is introduced. The SOR theory on determination of the optimal parameter is extended to the generalized method to include a wide class of matrices. The asymptotic rate of convergence of the new method is derived for Hermitian positive definite matrices using bounds of the eigenvalues of Jacobi matrices and numerical radius. Finally, numerical tests are presented to corroborate the obtained results.

Linear systems with signed solutions