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A tuple (Z 1 , ... , Z p) of matrices of size r × r is said to be a commuting extension of a tuple (A 1 , ... , A p) of matrices of size n × n if the Z i pairwise commute and each A i sits in the upper left corner of a block decomposition of Z i (here, r and n are two arbitrary integers with n < r). This notion was discovered and rediscovered in several contexts including algebraic complexity theory (in Strassen's work on tensor rank), in numerical analysis for the construction of cubature formulas and in quantum mechanics for the study of computational methods and the study of the so-called "quantum Zeno dynamics." Commuting extensions have also attracted the attention of the linear algebra community. In this paper we present 3 types of results: (i) Theorems on the uniqueness of commuting extensions for three matrices or more. (ii) Algorithms for the computation of commuting extensions of minimal size. These algorithms work under the same assumptions as our uniqueness theorems. They are applicable up to r = 4 n / 3 , and are apparently the first provably efficient algorithms for this problem applicable beyond r = n + 1. (iii) A genericity theorem showing that our algorithms and uniqueness theorems can be applied to a wide range of input matrices. [ABSTRACT FROM AUTHOR]

We give a generating function for the number of pairs of n × n matrices (A , B) over a finite field that are mutually annihilating, namely, A B = B A = 0. This generating function can be viewed as a singular analogue of a series considered by Cohen and Lenstra. We show that this generating function has a factorization that allows it to be meromorphically extended to the entire complex plane. We also use it to count pairs of mutually annihilating nilpotent matrices. This work is essentially a study of the motivic aspects about the variety of modules over C [ u , v ] / (u v) as well as the moduli stack of coherent sheaves over an algebraic curve with nodal singularities. [ABSTRACT FROM AUTHOR]

We consider the decidability of the membership problem for matrix-exponential semigroups: Given k ∈ N and square matrices A1, . . . ,Ak ,C, all of the same dimension and with real algebraic entries, decide whether C is contained in the semigroup generated by the matrix exponentials exp(Ait ), where i ∈ {1, . . . , k} and t ≥ 0. This problem can be seen as a continuous analog of Babai et al.’s and Cai et al.’s problem of solving multiplicative matrix equations and has applications to reachability analysis of linear hybrid automata and switching systems. Our main results are that the semigroup membership problem is undecidable in general, but decidable if we assume that A1, . . . ,Ak commute. The decidability proof is by reduction to a version of integer programming that has transcendental constants. We give a decision procedure for the latter using Baker’s theorem on linear forms in logarithms of algebraic numbers, among other tools. The undecidability result is shown by reduction from Hilbert’s Tenth Problem [ABSTRACT FROM AUTHOR]

This paper investigates the geometric properties of a special case of the two-sided system given by 2 × 2 tropical commuting constraints. Given a finite matrix A ∈ R 2 × 2 , the paper studies the extremals of the tropical polyhedral cone generated by the entries of matrices B such that A ⊗ B = B ⊗ A and proposes a criterion to test whether two 2 × 2 matrices commute in max linear algebra. [ABSTRACT FROM AUTHOR]

We give an Atiyah–Drinfel'd–Hitchin–Manin (ADHM) description of the Quot scheme of points Quot ℂ n (c , r) , of length c and rank r on affine spaces ℂ n which naturally extends both Baranovsky's representation of the punctual Quot scheme on a smooth surface and the Hilbert scheme of points on affine spaces ℂ n , described by the first author and M. Jardim. Using results on the variety of commuting matrices, and combining them with our construction, we prove new properties concerning irreducibility and reducedness of Quot ℂ n (c , r) and its punctual version Quot 𝕐 [ p ] (c , r) , where p is a fixed point on a smooth affine variety 𝕐. In this last case, we also study a connectedness result, for some special cases of higher r and c. [ABSTRACT FROM AUTHOR]

A tropical version of Stickel's key exchange protocol was suggested by Grigoriev and Shpilrain (2014) and successfully attacked by Kotov and Ushakov (2018). We suggest some modifications of this scheme that use commuting matrices in tropical algebra and discuss some possibilities of attacks on these new modifications. We suggest some simple heuristic attacks on one of our new protocols, and then we generalize the Kotov and Ushakov attack on tropical Stickel's protocol and discuss the application of that generalized attack to all our new protocols. [ABSTRACT FROM AUTHOR]

Given an n × n matrix A over C and an invariant subspace N , a straightforward formula constructs an n × n matrix N that commutes with A and has N = ker ⁡ N. For Q a matrix putting A into Jordan canonical form, J = Q − 1 A Q , we get N = Q − 1 M where M = ker(M) is an invariant subspace for J with M commuting with J. In the formula M = P Z T − 1 P t , the matrices Z and T are m × m and P is an n × m row selection matrix. If N is a marked subspace, m = n and Z is an n × n block diagonal matrix, and if N is not a marked subspace, then m > n and Z is an m × m near-diagonal block matrix. Strikingly, each block of Z is a monomial of a finite-dimensional backward shift. Each possible form of Z is easily arranged in a lattice structure isomorphic to and thereby displaying the complete invariant subspace lattice L (A) for A. [ABSTRACT FROM AUTHOR]

Commutative K-theory, a cohomology theory built from spaces of commuting matrices, has been explored in recent work of Adem, Gómez, Gritschacher, Lind and Tillman. In this article, we use unstable methods to construct explicit representatives for the real commutative K-theory classes on surfaces. These classes arise from commutative O ⁢ (2) {O(2)} -valued cocycles and are analyzed via the point-wise inversion operation on commutative cocycles. [ABSTRACT FROM AUTHOR]

Highlights • Sampling-type discrete fractional Fourier transforms (DFRFTs) and eigenvector decomposition-type DFRFTs are systematically analyzed and compared. • This work provides reasonable guidance for choosing a more appropriate DFRFT in different applications. • The differences among the DFRFT, discrete linear canonical transform (DLCT) and discrete simplified FRFT (DSFRFT) are analyzed both theoretically and experimentally. Abstract The fractional Fourier transform (FRFT) is a powerful tool for time-varying signal analysis. There exist various discrete fractional Fourier transforms (DFRFTs); in this paper, we systematically analyze and compare the main DFRFT types: sampling-type DFRFTs and eigenvector decomposition-type DFRFTs. First, for the existing sampling-type DFRFTs, we perform concrete analyses and comparisons of their applicable conditions and then establish their equivalence relationship. Then, for various eigenvector decomposition-type DFRFTs, their common mechanisms are extracted and thus they are effectively classified. In addition, as the extended version of DFRFTs, discrete counterparts of the linear canonical transform (LCT) and simplified FRFT (SFRFT) are summarized and classified. Our work is instructive for research about the choice of a more appropriate DFRFT in different applications, which is also supported by simulation experiments. Finally, for the DFRFT, DLCT and DSFRFT, two applications regarding detection for chirp signals and optical imaging are investigated to intuitively analyze their differences. [ABSTRACT FROM AUTHOR]

Abstract In 1961, Gerstenhaber proved the following theorem: if k is a field and X and Y are commuting d × d matrices with entries in k , then the unital k -algebra generated by these matrices has dimension at most d. The analog of this statement for four or more commuting matrices is false. The three matrix version remains open. We use commutative–algebraic techniques to prove that the three matrix analog of Gerstenhaber's theorem is true for some new classes of examples. In particular, we translate this three commuting matrix statement into an equivalent statement about certain maps between modules, and prove that this commutative–algebraic reformulation is true in special cases. We end with ideas for an inductive approach intended to handle the three matrix analog of Gerstenhaber's theorem more generally. [ABSTRACT FROM AUTHOR]

We prove that the algebraic set of pairs of matrices with a diagonal commutator over a field of positive prime characteristic, its irreducible components, and their intersection are F -pure when the size of matrices is equal to 3. Furthermore, we show that this algebraic set is reduced and the intersection of its irreducible components is irreducible in any characteristic for pairs of matrices of any size. In addition, we discuss various conjectures on the singularities of these algebraic sets and the system of parameters on the corresponding coordinate rings. [ABSTRACT FROM AUTHOR]

We reconsider the classification of all the factorizations of a matrix polynomial P as P = Q R with Q a matrix polynomial and R ( λ ) = λ T − S a regular matrix pencil. It is shown that the entire classification problem can be reduced to the simpler classification of factors R with commuting coefficients S , T . It is then shown that, for these commuting factors, S and T must satisfy a certain algebraic equation which we call the bisolvent equation. This extends the generalized Bézout theorem which associates monic factors λ I − S with solutions S of a solvent equation. In case P is regular, the classification of commuting pairs ( S , T ) of this type (up to left equivalence) is described in terms of enlarged standard pairs, following a well known approach. Under a non-derogatory generic condition on the roots of P , the number of such pairs associated with degree-minimal factorizations is finite, and admits explicit description in terms of Jordan chains. [ABSTRACT FROM AUTHOR]

We analyze tensors in C m ⊗ C m ⊗ C m satisfying Strassen's equations for border rank m . Results include: two purely geometric characterizations of the Coppersmith–Winograd tensor, a reduction to the study of symmetric tensors under a mild genericity hypothesis, and numerous additional equations and examples. This study is closely connected to the study of the variety of m -dimensional abelian subspaces of End ( C m ) and the subvariety consisting of the Zariski closure of the variety of maximal tori, called the variety of reductions. [ABSTRACT FROM AUTHOR]

We study tropical commuting matrices from two viewpoints: linear algebra and algebraic geometry. In classical linear algebra, there exist various criteria to test whether two square matrices commute. Similarly, in the realm of tropical linear algebra, we determine conditions for two tropical matrices that are Kleene stars to commute. Shifting to an algebro-geometric perspective, we explicitly compute the tropicalization of the classical variety of commuting matrices in dimensions 2 and 3. [ABSTRACT FROM AUTHOR]

For two n -by- n matrices A and B , it was known before that their numerical radii satisfy the inequality w ( A B ) ≤ 4 w ( A ) w ( B ) , and the equality is attained by the 2-by-2 matrices A = [ 0 1 0 0 ] and B = [ 0 0 1 0 ] . Moreover, the constant “4” here can be reduced to “2” if A and B commute, and the corresponding equality is attained by A = I 2 ⊗ [ 0 1 0 0 ] and B = [ 0 1 0 0 ] ⊗ I 2 . In this paper, we give a complete characterization of A and B for which the equality holds in each case. More precisely, it is shown that w ( A B ) = 4 w ( A ) w ( B ) (resp., w ( A B ) = 2 w ( A ) w ( B ) for commuting A and B ) if and only if either A or B is the zero matrix, or A and B are simultaneously unitarily similar to matrices of the form [ 0 a 0 0 ] ⊕ A ′ and [ 0 0 b 0 ] ⊕ B ′ (resp., ⊕ A ′ and ⊕ B ′ ) with w ( A ′ ) ≤ | a | / 2 and w ( B ′ ) ≤ | b | / 2 . An analogous characterization for the extremal equality for tensor products is also proven. For doubly commuting matrices, we use their unitary similarity model to obtain the corresponding result. For commuting 2-by-2 matrices A and B , we show that w ( A B ) = w ( A ) w ( B ) if and only if either A or B is a scalar matrix, or A and B are simultaneously unitarily similar to [ a 1 0 0 a 2 ] and [ b 1 0 0 b 2 ] with | a 1 | ≥ | a 2 | and | b 1 | ≥ | b 2 | . [ABSTRACT FROM AUTHOR]

Let F be a field. Any continuous function f from F n × n to F n × n satisfying x f ( x ) ≡ f ( x ) x can be written as f ( x ) ≡ α 0 ( x ) x 0 + … + α n − 1 ( x ) x n − 1 with α : F n × n → F n continuous on the set of non-derogatory matrices. Brešar and Šemrl showed this for F = C , and we give a proof valid over any topological field. For F ∈ { R , C } , we provide an example of a function f for which any appropriate α is discontinuous at every derogatory matrix. [ABSTRACT FROM AUTHOR]

Consider the space M n nor of square normal matrices X = ( x i j ) over R ∪ { − ∞ } , i.e., − ∞ ≤ x i j ≤ 0 and x i i = 0 . Endow M n nor with the tropical sum ⊕ and multiplication ⊙. Fix a real matrix A ∈ M n nor and consider the set Ω ( A ) of matrices in M n nor which commute with A . We prove that Ω ( A ) is a finite union of alcoved polytopes; in particular, Ω ( A ) is a finite union of convex sets. The set Ω A ( A ) of X such that A ⊙ X = X ⊙ A = A is also a finite union of alcoved polytopes. The same is true for the set Ω ′ ( A ) of X such that A ⊙ X = X ⊙ A = X . A topology is given to M n nor . Then, the set Ω A ( A ) is a neighborhood of the identity matrix I . If A is strictly normal, then Ω ′ ( A ) is a neighborhood of the zero matrix. In one case, Ω ( A ) is a neighborhood of A . We give an upper bound for the dimension of Ω ′ ( A ) . We explore the relationship between the polyhedral complexes span A , span X and span ( A X ) , when A and X commute. Two matrices, denoted A ̲ and A ¯ , arise from A , in connection with Ω ( A ) . The geometric meaning of them is given in detail, for one example. We produce examples of matrices which commute, in any dimension. [ABSTRACT FROM AUTHOR]

Queues with Markovian arrival and service processes, i.e., MAP/MAP/1 queues, have been useful in the analysis of computer and communication systems and different representations for their stationary sojourn time and queue length distribution have been derived. More specifically, the class of MAP/MAP/1 queues lies at the intersection of the class of QBD queues and the class of semi-Markovian queues. While QBD queues have a matrix exponential representation for their queue length and sojourn time distribution of orderNand N2, respectively, whereNis the size of the background continuous time Markov chain, the reverse is true for a semi-Markovian queue. As the class of MAP/MAP/1 queues lies at the intersection, both the queue length and sojourn time distribution of a MAP/MAP/1 queue has an orderNmatrix exponential representation. The aim of this article is to understand why the order N2distributions of the sojourn time of a QBD queue and the queue length of a semi-Markovian queue can be reduced to an orderNdistribution in the specific case of a MAP/MAP/1 queue. We show that the key observation exists in establishing the commutativity of some fundamental matrices involved in the analysis of the MAP/MAP/1 queue. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we introduce a new Jacobi–Davidson type eigenvalue solver for a set of commuting matrices, called JDCOMM, used for the global optimization of so-called Minkowski-norm dominated polynomials in several variables. The Stetter–Möller matrix method yields such a set of real non-symmetric commuting matrices since it reformulates the optimization problem as an eigenvalue problem. A drawback of this approach is that the matrix most relevant for computing the global optimum of the polynomial under investigation is usually large and only moderately sparse. However, the other matrices are generally much sparser and have the same eigenvectors because of the commutativity. This fact is used to design the JDCOMM method for this problem: the most relevant matrix is used only in the outer loop and the sparser matrices are exploited in the solution of the correction equation in the inner loop to greatly improve the efficiency of the method. Some numerical examples demonstrate that the method proposed in this paper is more efficient than approaches that work on the main matrix (standard Jacobi–Davidson and implicitly restarted Arnoldi), as well as conventional solvers for computing the global optimum, i.e., SOSTOOLS, GloptiPoly, and PHCpack. [Copyright &y& Elsevier]

Abstract: This is essentially the text of my talk on Fiedler–Pták scaling in max algebra delivered in the invited minisymposium in honor of Miroslav Fiedler at the 17th ILAS Conference in Braunschweig, Germany. [Copyright &y& Elsevier]

We study the structure of the nilpotent commutator 𝒩 B of a nilpotent matrix B. We show that 𝒩 B intersects all nilpotent orbits for conjugation if and only if B is a square-zero matrix. We describe nonempty intersections of 𝒩 B with nilpotent orbits in the case the n × n matrix B has rank n − 2. Moreover, we give some results concerning the inverse image of the map taking B to the maximal nilpotent orbit intersecting 𝒩 B . [ABSTRACT FROM AUTHOR]

We present an extension of the QR method to simultaneously compute the joint eigenvalues of a finite family of commuting matrices. The problem is motivated by the task of finding solutions of a polynomial system. Several examples are included. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies commuting matrices in max algebra and nonnegative linear algebra. Our starting point is the existence of a common eigenvector which directly leads to max analogs and nonnegative analogs of some classical results for complex matrices. We also investigate Frobenius normal forms of commuting matrices, particularly when the Perron roots of the components are distinct. For the case of max algebra, we show how the intersection of eigencones of commuting matrices can be described and we consider connections with Boolean algebra which enables us to prove that two commuting irreducible matrices in max algebra have a common eigennode. [Copyright &y& Elsevier]

In signal processing, coprime pairs of integer matrices have an important role in the study of multidimensional multirate systems, and in multidimensional array processing. In this paper a general theorem for coprimality of pairs of matrices is first presented, based on the minors of an associated composite matrix. First, a necessary and sufficient set of conditions for coprimality is presented based on minors. This result applies to matrices of any size. Several useful consequences of the result are discussed, and it is first applied to the 2\,\times\,2 case, including special cases such as Toeplitz matrices, commuting Toeplitz matrices, circulants, and skew circulants. The condition under which a 2\,\times\,2 integer matrix and its ajdugate are coprime is also derived. The minor based result is then applied extensively for the 3\,\times\,3 case to derive new coprimality conditions for several matrix pairs. In particular, adjugates of matrices are considered in detail because adjugates always commute with the parent matrices. The so-called generalized 3\,\times\,3 circulant matrix and its adjugate are then analyzed in detail. Generalized circulants also form a commuting family. Special cases such as 3\,\times\,3 circulant pairs and skew circulant pairs are also elaborated. [ABSTRACT FROM PUBLISHER]

Commuting coprime integer matrices arise in signal processing in a number of contexts. This paper develops two ways, one nonlinear and the other linear, to generate new commuting coprime pairs from known pairs. The first method is based on computing powers of the initial pair of matrices, and the second method is based on appropriate types of linear combinations of the initial pair of matrices. This enriches the already known families of coprime matrices reported in recent literature. Several properties of the newly generated coprime pairs are also addressed. [ABSTRACT FROM PUBLISHER]

Commuting coprime pairs of integer matrices have been of interest in multidimensional multirate systems, and more recently in array processing. In multirate systems they arise, for example, in the design of interchangeable cascades of decimator and expander matrices. In array processing they arise in the construction of dense coarrays from sparse sensors located on a pair of lattices. For the important case of two dimensional signals, these matrices have size 2 \times 2. In this paper the condition for coprimality is derived for several classes of 2 \times 2 integer matrices, namely circulant, skew-circulant, and triangular families. The first two are also commuting families. For each class, the special case of adjugate pairs, which automatically commute, is also elaborated. It is also shown that the problem of testing coprimality of two 2 \times 2 matrices is equvialent to testing coprimality of a pair of triangular matrices, which can be done almost by inspection. Also considered is the case of 3 \times 3 triangular matrices and their adjugates, which have potential applications in three dimensional signal processing. [ABSTRACT FROM AUTHOR]

Abstract: In order to define the discrete fractional Fourier transform, Hermite Gauss-like eigenvectors are needed and one way of extracting these eigenvectors is to employ DFT commuting matrices. Recently, Pei et al. exploited the idea of obtaining higher order DFT-commuting matrices, which was introduced by Candan previously. The upper bound of O(h 2k ) approximation to N×N commuting matrix is in Candan''s work and Pei et al. improved the proximity by removing this upper bound at the expense of higher computational cost. In this paper, we derive an exact closed form expression of infinite-order Taylor series approximation to discrete second derivative operator and employ it in the definition of excellent DFT commuting matrices. We show that in the limit this Taylor series expansion converges to a trigonometric function of second-order differentiating matrix. The commuting matrices possess eigenvectors that are closer to the samples of Hermite–Gaussian eigenfunctions of DFT better than any other methods in the literature with no additional computational cost. [Copyright &y& Elsevier]