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We study matrix identities involving multiplication and unary operations such as transposition or Moore-Penrose inversion. We prove that in many cases such identities admit no finite basis.
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In matrix theory, a well established relation $(AB)^{T}=B^{T}A^{T}$ holds for any two matrices $A$ and $B$ for which the product $AB$ is defined. Here $T$ denote the usual transposition. In this work, we explore the possibility of deriving the matrix equality $(AB)^{Gamma}=A^{Gamma}B^{Gamma}$ for any $4 times 4$ matrices $A$ and $B$, where $Gamma$ denote the partial transposition. We found that, in general, $(AB)^{Gamma} eq A^{Gamma}B^{Gamma}$ holds for $4 times 4$ matrices $A$ and $B$ but there exist particular set of $4 times 4$ matrices for which $(AB)^{Gamma}= A^{Gamma}B^{Gamma}$ holds. We have exploited this matrix equality to investigate the separability problem. Since it is possible to decompose the density matrices $rho$ into two positive semi-definite matrices $A$ and $B$ so we are able to derive the separability condition for $rho$ when $rho^{Gamma}=(AB)^{Gamma}=A^{Gamma}B^{Gamma}$ holds. Due to the non-uniqueness property of the decomposition of the density matrix into two positive semi-definte matrices $A$ and $B$, there is a possibility to generalise the matrix equality for density matrices lives in higher dimension. These results may help in studying the separability problem for higher dimensional and multipartite system.
The Cohn-Umans group-theoretic approach to matrix multiplication suggests embedding matrix multiplication into group algebra multiplication, and bounding $omega$ in terms of the representation theory of the host group. This framework is general enough to capture the best known upper bounds on $omega$ and is conjectured to be powerful enough to prove $omega = 2$, although finding a suitable group and constructing such an embedding has remained elusive. Recently it was shown, by a generalization of the proof of the Cap Set Conjecture, that abelian groups of bounded exponent cannot prove $omega = 2$ in this framework, which ruled out a family of potential constructions in the literature. In this paper we study nonabelian groups as potential hosts for an embedding. We prove two main results: (1) We show that a large class of nonabelian groups---nilpotent groups of bounded exponent satisfying a mild additional condition---cannot prove $omega = 2$ in this framework. We do this by showing that the shrinkage rate of powers of the augmentation ideal is similar to the shrinkage rate of the number of functions over $(mathbb{Z}/pmathbb{Z})^n$ that are degree $d$ polynomials; our proof technique can be seen as a generalization of the polynomial method used to resolve the Cap Set Conjecture. (2) We show that symmetric groups $S_n$ cannot prove nontrivial bounds on $omega$ when the embedding is via three Young subgroups---subgroups of the form $S_{k_1} times S_{k_2} times dotsb times S_{k_ell}$---which is a natural strategy that includes all known constructions in $S_n$. By developing techniques for negative results in this paper, we hope to catalyze a fruitful interplay between the search for constructions proving bounds on $omega$ and methods for ruling them out.
We first establish the result that the Narayana polynomials can be represented as the integrals of the Legendre polynomials. Then we represent the Catalan numbers in terms of the Narayana polynomials by three different identities. We give three different proofs for these identities, namely, two algebraic proofs and one combinatorial proof. Some applications are also given which lead to many known and new identities.
We propose to store several integers modulo a small prime into a single machine word. Modular addition is performed by addition and possibly subtraction of a word containing several times the modulo. Modular Multiplication is not directly accessible but modular dot product can be performed by an integer multiplication by the reverse integer. Modular multiplication by a word containing a single residue is a also possible. Therefore matrix multiplication can be performed on such a compressed storage. We here give bounds on the sizes of primes and matrices for which such a compression is possible. We also explicit the details of the required compressed arithmetic routines.
Two intimately related new classes of games are introduced and studied: entropy games (EGs) and matrix multiplication games (MMGs). An EG is played on a finite arena by two-and-a-half players: Despot, Tribune and the non-deterministic People. Despot wants to make the set of possible Peoples behaviors as small as possible, while Tribune wants to make it as large as possible.An MMG is played by two players that alternately write matrices from some predefined finite sets. One wants to maximize the growth rate of the product, and the other to minimize it. We show that in general MMGs are undecidable in quite a strong sense.On the positive side, EGs correspond to a subclass of MMGs, and we prove that such MMGs and EGs are determined, and that the optimal strategies are simple. The complexity of solving such games is in NP&coNP.
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