DOCSLIB.ORG

The Transitive Closures of Matrices Over Idempotent Semirings and Its Applications*

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

International Conference on Advanced Computer Science and Electronics Information (ICACSEI 2013) The Transitive Closures of Matrices over Idempotent Semirings and its Applications* Wang Zhixi1, Wang Yana1, He Yong1, Hu Binliang2 1 School of Computer Science and Engineering, Hunan University of Science and Technology, Xiangtan 411201, Hunan, China 2Colleges of Electromechanical Engineering, Hunan University of Science and Technology, Xiangtan 411201, Hunan, China [email protected], [email protected], [email protected], [email protected] Abstract - The Algebraic Path Problem is an important issue method has been improved from every angle and its with a broad background. Consider the problem to solve the application has a gradual expansion from some special algebraic path problem can be concluded to find the Kleene closure semirings first to general cases, and the computing power of the adjacency matrix, an algorithm of time complexity O() n 3 to with the introduction of advanced technologies such as find the transitive closure of matrices over idempotent semiring is parallel computing has been greatly improved [7-14]. constructed, as well as the conditions applicable to it are provided. However, the semirings applied to Gauβ- Jordan elimination Compared with the Gauβ-Jordan elimination, this algorithm could method always required completeness and closeness. extend the applicable range to certain semirings which do not have Note that the Kleene Closures of matrices over completeness and closeness, Thus, it has a wide application since it Idempotent semirings are easily obtained by their Transitive can provide a new method to solve the algebraic path problem over Closures. This paper constructs an algorithm to find the idempotent semirings which do not have completeness and closeness. Transitive Closures of matrices over idempotent semiring and Index Terms - Idempotent semiring, Transitive Closures, provides conditions applicable to it. Actually, the algorithm Transitive_Closure Algorithm, Algebraic Path Problem. extends the applicable range of Gauβ-Jordan elimination method to certain semirings which do not have completeness I. Introduction and closeness. The Algebraic Path Problem, which is to perform a II. Preliminary special unary operation, called closure, on a square matrix over semirings based on semirings-algebraic theory, is an Let R be a non-empty set defined the addition and important issue with a broad background. This problem the multiplication . If (,)R is a commutative monoid, and generalizes some matrix problems(such as computing the (,)R is a semigroup, moreover, multiplication satisfies inverse of a real-valued matrix), graph problems(e.g. transitive closure, shortest path, the stochastic communication the distributive law to addition , then the algebraic system network problem), and regular language problems(e.g. the RR(,,) is called a semiring. Usually, the product a b proof of the correspondence between regular expressions and of elements a , b over semiring R is denoted byab . Denote finite automata). Algebraic path problem and its algorithms that a(m ) a a 2 am for the element a of R and positive have a wide application in the field of network engineering, image processing and pattern recognition. integer m . In 1956, Kleene proved that every regular language can Assume that R is a semiring. If 1R is the identity be represented by a regular expression. Actually, it was the element of (,)R , then we say that 1 is the identity element first algorithm for the algebraic path problem[1]. And in of R . If 0R is the identity element of (,)R and the zero 1962, Warshall algorithm for computing the transitive closure and Floyd algorithm for shortest path problem were element of(,)R , then we say 0 is the zero element of R . respectively proposed[2-3]. Later, Gaussian elimination for Define the relation over semiring R as computing the inverse of matrices was also presented[4-5]. a b a b a . We call R an idempotent semiring if for any Warshall algorithms, Floyd algorithms and Gaussian a R , then a a . It is easy to prove that the relation over elimination method are classical algorithms for these three the idempotent semirings is a partial order relation. The problems respectively [2-4]. Until 1971, Carré discovered the idempotent semirings have a significant application in similarities between these algorithms and proposed their solving optimization problems and the classification of general form Gauβ-Jordan elimination for solving them based regular languages, c.f. [14-16]. on semirings algebra theory. The algorithm for solving algebraic path problem boiled down to find the Kleene For the element a of R , let (a ) { x R x a, x xx} , closure of adjacency matrix, i.e., the summation of all non- we call that at is the transitive closure of a if at () a and negative integer powers of the adjacency matrix[6]. Since t then, with an in-depth study, Gauβ-Jordan elimination for any x()A , x a . * This work is supported by National Nature Science Foundation of China (No:61272196), Hunan Provincial Natural Science Foundation of China ( No: 10JJ3067), and Scientific Research Fund of Hunan Provincial Education Department (No: 11C0534). © 2013. The authors - Published by Atlantis Press 174 Similar to the numerical matrices, we can define The Analysis of The Complexity of Algorithm. matrices and their addition and multiplication over semiring The algorithm firstly is to compute the matrices A[]n , R . Denote the element (,)i j of A over R by Aij , and then is to check the applicable conditions of the algorithm. Taken the assignment, addition, multiplication and matrix that the element (,)i j is a over R by ()a . The set ij ij comparison operations of the elements of the semiring as a of all n square matrix over R is recorded as MRn() .Then basic operation respectively, we can conclude that the first 2 part needs 3n ( n 1) times of basic operations, and the second MRn() with regard to matrix addition and multiplication also forms a semiring called an n square matrix semiring over part takes up at most 2n 3 times of the basic operations. 3 R . The relation on MRn() is defined as: ABAB ij ij Hence, the time complexity of the algorithm is O() n , which has the same order with the time complexity of the Floyd for 1i , j n . Obviously, MRn() is idempotent if and only if algorithm and the Gauβ-Jordan algorithm, c.f. [4], [12]. R is idempotent. For the n square matrix A over R and positive integer m , we set AAAA(m ) 2 m , and denote IV. Proof Of The Transitive_Closure Algorithm m m the element (,)i j of A by Aij , and also the element In this section, we assume that R is an idempotent ()m ()m semiring, n is a given positive integer, and A()() aij M n R . (,)i j of A by Aij . The basic properties of matrices We need Lemma 2 to Lemma 4 as follows to prove the over semirings and the details of semirings can be found in Transitive_Closure algorithm, i.e., the Theorem 1. literature [15]. Lemma 2 Let a,,,, b c d x R with 1i m . Then the following For the n square matrix A over R and the non- i [][]k k condition holds: negative integer k with kn , we define matrices AA ij m a x a x i1, 2, , m recursively as follows: (1) i if and only if i , for . i1 A , k0 ij (2) If there exists an integer i such that xi a , then AA[k ] [ k 1] , i k or j k ij ij m A[k 1] A [ k 1] A [ k 1] , 1 k n and i k and j k ij ik kj xi a . i1 III. The Analysis Of The Complexity Of Transitive_Closure (3) If a b , cd , then acbd . Algorithm Proof. (1) Assume that a xi with i1, 2, , m , then The following theorem is the main conclusion of this m m m article, and we will consider the proof of this matter in next a xi () a xi a a . section. i1 i 1 i1 m Theorem 1. Let R is an idempotent semiring, AMRn() , Conversely, assume that a xi , then for each xi , we i1 t[] n [][][]n n n then AA if and only if AAAij ik kj for 1i , j , k n . m m have a xi a xi x i a xi a . Theorem 1 provides an algorithm to compute the i1 i1 transitive closures of the n square matrix over idempotent (2) Assume that there exists an integer i such that semiring, the algorithm can be described by the pseudo codes x a . Further assume that i1 , then as follows: i m m m Transitive_Closure Algorithm a xi () a x1 xi x i . i1 i2 i 1 Function Transitive_Closure(Matrix A, Matrix W) (3) Assume that a b , c d , then n = A.rows, W = A ac bd ( a b )( c d ) bd '' n is the order of the matrix, W is the result matrix '' First part: Calculate the matrix W, i.e., the matrix A[n] ac ad bc bd bd For k = 1 To n ac ad bc bd . For i = 1 To n, j = 1 To n Do '' Calculate the matrix A[k] (a b )( c d ) If i = k Or j = k Then Continue For ac W(i, j) = W(i, j) + W(i, k) * W(k, j)) Lemma 3. Assume a R , x() a , then the following Next Next k conditions holds: '' Second part: Check the applicable condition of the algorithm (max(r , p q )) For i = 1 To n, j = 1 To n, k = 1 To n Do (1) a()()()r a p a q a , where r,, p q are all positive '' Check whether W <= W * W integers.
Recommended publications
  • Relations 21

    Relations 21

    2. CLOSURE OF RELATIONS 21 2. Closure of Relations 2.1. Definition of the Closure of Relations. Definition 2.1.1. Given a relation R on a set A and a property P of relations, the closure of R with respect to property P , denoted ClP (R), is smallest relation on A that contains R and has property P . That is, ClP (R) is the relation obtained by adding the minimum number of ordered pairs to R necessary to obtain property P . Discussion To say that ClP (R) is the “smallest” relation on A containing R and having property P means that • R ⊆ ClP (R), • ClP (R) has property P , and • if S is another relation with property P and R ⊆ S, then ClP (R) ⊆ S. The following theorem gives an equivalent way to define the closure of a relation. \ Theorem 2.1.1. If R is a relation on a set A, then ClP (R) = S, where S∈P P = {S|R ⊆ S and S has property P }. Exercise 2.1.1. Prove Theorem 2.1.1. [Recall that one way to show two sets, A and B, are equal is to show A ⊆ B and B ⊆ A.] 2.2. Reflexive Closure. Definition 2.2.1. Let A be a set and let ∆ = {(x, x)|x ∈ A}. ∆ is called the diagonal relation on A. Theorem 2.2.1. Let R be a relation on A. The reflexive closure of R, denoted r(R), is the relation R ∪ ∆. Proof. Clearly, R ∪ ∆ is reflexive, since (a, a) ∈ ∆ ⊆ R ∪ ∆ for every a ∈ A.
  • Scalability of Reflexive-Transitive Closure Tables As Hierarchical Data Solutions

    Scalability of Reflexive-Transitive Closure Tables As Hierarchical Data Solutions

    Scalability of Reflexive- Transitive Closure Tables as Hierarchical Data Solutions Analysis via SQL Server 2008 R2 Brandon Berry, Engineer 12/8/2011 Reflexive-Transitive Closure (RTC) tables, when used to persist hierarchical data, present many advantages over alternative persistence strategies such as path enumeration. Such advantages include, but are not limited to, referential integrity and a simple structure for data queries. However, RTC tables grow exponentially and present a concern with scaling both operational performance of the RTC table and volume of the RTC data. Discovering these practical performance boundaries involves understanding the growth patterns of reflexive and transitive binary relations and observing sample data for large hierarchical models. Table of Contents 1 Introduction ......................................................................................................................................... 2 2 Reflexive-Transitive Closure Properties ................................................................................................. 3 2.1 Reflexivity ...................................................................................................................................... 3 2.2 Transitivity ..................................................................................................................................... 3 2.3 Closure .......................................................................................................................................... 4 3 Scalability
  • MATH 213 Chapter 9: Relations

    MATH 213 Chapter 9: Relations

    MATH 213 Chapter 9: Relations Dr. Eric Bancroft Fall 2013 9.1 - Relations Definition 1 (Relation). Let A and B be sets. A binary relation from A to B is a subset R ⊆ A×B, i.e., R is a set of ordered pairs where the first element from each pair is from A and the second element is from B. If (a; b) 2 R then we write a R b or a ∼ b (read \a relates/is related to b [by R]"). If (a; b) 2= R, then we write a R6 b or a b. We can represent relations graphically or with a chart in addition to a set description. Example 1. A = f0; 1; 2g;B = f1; 2; 3g;R = f(1; 1); (2; 1); (2; 2)g Example 2. (a) \Parent" (b) 8x; y 2 Z; x R y () x2 + y2 = 8 (c) A = f0; 1; 2g;B = f1; 2; 3g; a R b () a + b ≥ 3 1 Note: All functions are relations, but not all relations are functions. Definition 2. If A is a set, then a relation on A is a relation from A to A. Example 3. How many relations are there on a set with. (a) two elements? (b) n elements? (c) 14 elements? Properties of Relations Definition 3 (Reflexive). A relation R on a set A is said to be reflexive if and only if a R a for all a 2 A. Definition 4 (Symmetric). A relation R on a set A is said to be symmetric if and only if a R b =) b R a for all a; b 2 A.
  • Properties of Binary Transitive Closure Logics Over Trees S K

    Properties of Binary Transitive Closure Logics Over Trees S K

    8 Properties of binary transitive closure logics over trees S K Abstract Binary transitive closure logic (FO∗ for short) is the extension of first-order predicate logic by a transitive closure operator of binary relations. Deterministic binary transitive closure logic (FOD∗) is the restriction of FO∗ to deterministic transitive closures. It is known that these logics are more powerful than FO on arbitrary structures and on finite ordered trees. It is also known that they are at most as powerful as monadic second-order logic (MSO) on arbitrary structures and on finite trees. We will study the expressive power of FO∗ and FOD∗ on trees to show that several MSO properties can be expressed in FOD∗ (and hence FO∗). The following results will be shown. A linear order can be defined on the nodes of a tree. The class EVEN of trees with an even number of nodes can be defined. On arbitrary structures with a tree signature, the classes of trees and finite trees can be defined. There is a tree language definable in FOD∗ that cannot be recognized by any tree walking automaton. FO∗ is strictly more powerful than tree walking automata. These results imply that FOD∗ and FO∗ are neither compact nor do they have the L¨owenheim-Skolem-Upward property. Keywords B 8.1 Introduction The question aboutthe best suited logic for describing tree properties or defin- ing tree languages is an important one for model theoretic syntax as well as for querying treebanks. Model theoretic syntax is a research program in mathematical linguistics concerned with studying the descriptive complex- Proceedings of FG-2006.
  • Descriptive Complexity: a Logician's Approach to Computation

    Descriptive Complexity: a Logician's Approach to Computation

    Descriptive Complexity a Logicians Approach to Computation Neil Immerman Computer Science Dept University of Massachusetts Amherst MA immermancsumassedu App eared in Notices of the American Mathematical Society A basic issue in computer science is the complexity of problems If one is doing a calculation once on a mediumsized input the simplest algorithm may b e the b est metho d to use even if it is not the fastest However when one has a subproblem that will havetobesolved millions of times optimization is imp ortant A fundamental issue in theoretical computer science is the computational complexity of problems Howmuch time and howmuch memory space is needed to solve a particular problem Here are a few examples of such problems Reachability Given a directed graph and two sp ecied p oints s t determine if there is a path from s to t A simple lineartime algorithm marks s and then continues to mark every vertex at the head of an edge whose tail is marked When no more vertices can b e marked t is reachable from s i it has b een marked Mintriangulation Given a p olygon in the plane and a length L determine if there is a triangulation of the p olygon of total length less than or equal to L Even though there are exp onentially many p ossible triangulations a dynamic programming algorithm can nd an optimal one in O n steps ThreeColorability Given an undirected graph determine whether its vertices can b e colored using three colors with no two adjacentvertices having the same color Again there are exp onentially many p ossibilities
  • Relations April 4

    Relations April 4

    math 55 - Relations April 4 Relations Let A and B be two sets. A (binary) relation on A and B is a subset R ⊂ A × B. We often write aRb to mean (a; b) 2 R. The following are properties of relations on a set S (where above A = S and B = S are taken to be the same set S): 1. Reflexive: (a; a) 2 R for all a 2 S. 2. Symmetric: (a; b) 2 R () (b; a) 2 R for all a; b 2 S. 3. Antisymmetric: (a; b) 2 R and (b; a) 2 R =) a = b. 4. Transitive: (a; b) 2 R and (b; c) 2 R =) (a; c) 2 R for all a; b; c 2 S. Exercises For each of the following relations, which of the above properties do they have? 1. Let R be the relation on Z+ defined by R = f(a; b) j a divides bg. Reflexive, antisymmetric, and transitive 2. Let R be the relation on Z defined by R = f(a; b) j a ≡ b (mod 33)g. Reflexive, symmetric, and transitive 3. Let R be the relation on R defined by R = f(a; b) j a < bg. Symmetric and transitive 4. Let S be the set of all convergent sequences of rational numbers. Let R be the relation on S defined by f(a; b) j lim a = lim bg. Reflexive, symmetric, transitive 5. Let P be the set of all propositional statements. Let R be the relation on P defined by R = f(a; b) j a ! b is trueg.
  • A Transitive Closure Algorithm for Test Generation

    A Transitive Closure Algorithm for Test Generation

    A Transitive Closure Algorithm for Test Generation Srimat T. Chakradhar, Member, IEEE, Vishwani D. Agrawal, Fellow, IEEE, and Steven G. Rothweiler Abstract-We present a transitive closure based test genera- Cox use reduction lists to quickly determine necessary as- tion algorithm. A test is obtained by determining signal values signments [5]. that satisfy a Boolean equation derived from the neural net- These algorithms solve two main problems: (1) they work model of the circuit incorporating necessary conditions for fault activation and path sensitization. The algorithm is a determine logical consequences of a partial set of signal sequence of two main steps that are repeatedly executed: tran- assignments and (2) they determine the order in which sitive closure computation and decision-making. To compute decisions on fixing signals should be made. the transitive closure, we start with an implication graph whose In spite of making extensive use of the circuit structure vertices are labeled as the true and false states of all signals. A directed edge (x, y) in this graph represents the controlling in- and function, most algorithms have several shortcomings. fluence of the true state of signal x on the true state of signal y First, they do not guarantee the identification of all logi- that are connected through a wire or a gate. Since the impli- cal consequences of a partial set of signal assignments. In cation graph only includes local pairwise (or binary) relations, other words, local conditions are easier to identify than it is a partial representation of the netlist. Higher-order signal the global ones.
  • Accelerating Transitive Closure of Large-Scale Sparse Graphs

    Accelerating Transitive Closure of Large-Scale Sparse Graphs

    New Jersey Institute of Technology Digital Commons @ NJIT Theses Electronic Theses and Dissertations 12-31-2020 Accelerating transitive closure of large-scale sparse graphs Sanyamee Milindkumar Patel New Jersey Institute of Technology Follow this and additional works at: https://digitalcommons.njit.edu/theses Part of the Computer Sciences Commons Recommended Citation Patel, Sanyamee Milindkumar, "Accelerating transitive closure of large-scale sparse graphs" (2020). Theses. 1806. https://digitalcommons.njit.edu/theses/1806 This Thesis is brought to you for free and open access by the Electronic Theses and Dissertations at Digital Commons @ NJIT. It has been accepted for inclusion in Theses by an authorized administrator of Digital Commons @ NJIT. For more information, please contact [email protected]. Copyright Warning & Restrictions The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted material. Under certain conditions specified in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these specified conditions is that the photocopy or reproduction is not to be “used for any purpose other than private study, scholarship, or research.” If a, user makes a request for, or later uses, a photocopy or reproduction for purposes in excess of “fair use” that user may be liable for copyright infringement, This institution reserves the right to refuse to accept a copying order if, in its judgment, fulfillment
  • Efficient Transitive Closure Algorithms

    Efficient Transitive Closure Algorithms

    Efficient Transitive Closure Algorithms Yards E. Ioannidis t Raghu Rantakrishnan ComputerSciences Department Universityof Wisconsin Madison.WJ 53706 Abstract closure (e.g. to find the shortestpaths between pairs of We havedeveloped some efficient algorithms for nodes) since it loses path information by merging all computing the transitive closure of a directed graph. nodes in a strongly connectedcomponent. Only the This paper presentsthe algorithmsfor the problem of Seminaive algorithm computes selection queries reachability. The algorithms,however, can be adapted efficiently. Thus, if we wish to find all nodesreachable to deal with path computationsand a signitkantJy from a given node, or to find the longest path from a broaderclass of queriesbased on onesided recursions. given node, with the exceptionof the Seminaivealgo- We analyze these algorithms and compare them to ritbm, we must essentiallycompute the entire transitive algorithms in the literature. The resulti indicate that closure(or find the longestpath from every node in the thesealgorithms, in addition to their ability to deal with graph)first and then perform a selection. queries that am generakations of transitive closure, We presentnew algorithmsbased on depth-first also perform very efficiently, in particular, in the con- searchand a schemeof marking nodes (to record ear- text of a dish-baseddatabase environment. lier computation implicitly) that computes transitive closure efficiently. They can also be adaptedto deal 1. Introduction with selection queries ‘and path computations Severaltransitive closure algoritluus have been efficiently. Jn particular, in the context of databasesan presentedin the literature.These include the Warshall importantconsideration is J/O cost, since it is expected and Warren algorithms, which use a bit-matrix that relations will not fit in main memory.
  • Block Matrices in Linear Algebra

    Block Matrices in Linear Algebra

    PRIMUS Problems, Resources, and Issues in Mathematics Undergraduate Studies ISSN: 1051-1970 (Print) 1935-4053 (Online) Journal homepage: https://www.tandfonline.com/loi/upri20 Block Matrices in Linear Algebra Stephan Ramon Garcia & Roger A. Horn To cite this article: Stephan Ramon Garcia & Roger A. Horn (2020) Block Matrices in Linear Algebra, PRIMUS, 30:3, 285-306, DOI: 10.1080/10511970.2019.1567214 To link to this article: https://doi.org/10.1080/10511970.2019.1567214 Accepted author version posted online: 05 Feb 2019. Published online: 13 May 2019. Submit your article to this journal Article views: 86 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=upri20 PRIMUS, 30(3): 285–306, 2020 Copyright # Taylor & Francis Group, LLC ISSN: 1051-1970 print / 1935-4053 online DOI: 10.1080/10511970.2019.1567214 Block Matrices in Linear Algebra Stephan Ramon Garcia and Roger A. Horn Abstract: Linear algebra is best done with block matrices. As evidence in sup- port of this thesis, we present numerous examples suitable for classroom presentation. Keywords: Matrix, matrix multiplication, block matrix, Kronecker product, rank, eigenvalues 1. INTRODUCTION This paper is addressed to instructors of a first course in linear algebra, who need not be specialists in the field. We aim to convince the reader that linear algebra is best done with block matrices. In particular, flexible thinking about the process of matrix multiplication can reveal concise proofs of important theorems and expose new results. Viewing linear algebra from a block-matrix perspective gives an instructor access to use- ful techniques, exercises, and examples.
  • CDM Relations

    CDM Relations

    CDM 1 Relations Relations Operations and Properties Orders Klaus Sutner Carnegie Mellon University Equivalence Relations 30-relations 2017/12/15 23:22 Closures Relations 3 Binary Relations 4 We have seen how to express general concepts (or properties) as sets: we form the set of all objects that “fall under” the concept (in Frege’s Let’s focus on the binary case where two objects are associated, though terminology). Thus we can form the set of natural numbers, of primes, of not necessarily from the same domain. reals, of continuous functions, of stacks, of syntactically correct The unifying characteristic is that we have some property (attribute, C-programs and so on. quality) that can hold or fail to hold of any two objects from the Another fundamental idea is to consider relationships between two or appropriate domains. more objects. Here are some typical examples: Standard notation: for a relation P and suitable objects a and b write P (a, b) for the assertion that P holds on a and b. divisibility relation on natural numbers, Thus we can associate a truth value with P (a, b): the assertion holds if a less-than relation on integers, and b are indeed related by P , and is false otherwise. greater-than relation on rational numbers, For example, if P denotes the divisibility relation on the integers then the “attends course” relation for students and courses, P (3, 9) holds whereas P (3, 10) is false. the “is prerequisite” relation for courses, Later we will be mostly interested in relations where we can effectively the “is a parent of” relation for humans, determine whether P (a, b) holds, but for the time being we will consider the “terminates on input and produces output” relation for programs the general case.
  • Matrix Theory States Ecd Eef = X (D = (3) E)Ecf

    Matrix Theory States Ecd Eef = X (D = (3) E)Ecf

    Tensor spaces { the basics S. Gill Williamson Abstract We present the basic concepts of tensor products of vectors spaces, ex- ploiting the special properties of vector spaces as opposed to more gen- eral modules. Introduction (1), Basic multilinear algebra (2), Tensor products of vector spaces (3), Tensor products of matrices (4), Inner products on tensor spaces (5), Direct sums and tensor products (6), Background concepts and notation (7), Discussion and acknowledge- ments (8). This material draws upon [Mar73] and relates to subsequent work [Mar75]. 1. Introduction a We start with an example. Let x = ( b ) 2 M2;1 and y = ( c d ) 2 M1;2 where Mm;n denotes the m × n matrices over the real numbers, R. The set of matrices Mm;n forms a vector space under matrix addition and multiplication by real numbers, R. The dimension of this vector space is mn. We write dim(Mm;n) = mn: arXiv:1510.02428v1 [math.AC] 8 Oct 2015 ac ad Define a function ν(x; y) = xy = bc bd 2 M2;2 (matrix product of x and y). The function ν has domain V1 × V2 where V1 = M2;1 and V2 = M1;2 are vector spaces of dimension, dim(Vi) = 2, i = 1; 2. The range of ν is the vector space P = M2;2 which has dim(P ) = 4: The function ν is bilinear in the following sense: (1.1) ν(r1x1 + r2x2; y) = r1ν(x1; y) + r2ν(x2; y) (1.2) ν(x; r1y1 + r2y2) = r1ν(x; y1) + r2ν(x; y2) for any r1; r2 2 R; x; x1; x2 2 V1; and y; y1; y2 2 V2: We denote the set of all such bilinear functions by M(V1;V2 : P ).