DOCSLIB.ORG

What Is a Tensor?

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
What Is a Tensor? WHAT ISA TENSOR? As we have seen in class, there are a lot of different ways to understand a tensor. The most fundamental way is to think of a tensor as a vector in a space constructed from taking the tensor product. Let V; W be vector spaces. The tensor product V ⊗ W is the set of elements n X civi ⊗ wi i=1 such that the following holds: (1) (v1 + v2) ⊗ w = v1 ⊗ w + v2 ⊗ w (2) v ⊗ (w1 + w2) = v ⊗ w1 + v ⊗ w2 (3) (cv) ⊗ w = c(v ⊗ w) = v ⊗ (cw) for all vi 2 V and wi 2 W . Given two vectors v; w, we can treat v ⊗ w as the Kronecker product: (v1; v2; v3) ⊗ (w1; w2) = (v1w1; v1w2; v2w1; v2w2; v3w1; v3w2) 2 3 Exercise 1. Show that the vector η below is an element of R ⊗ R but that there does not exist vectors 2 3 v 2 R ; w 2 R such that v ⊗ w = η. η = (0; 1; 1; 2; 0; 1) Solution 1. Suppose the contrary. Because the first entry is zero and R is a field, then either v1 = 0 or w1 = 0. The second entry indicates that v1 6= 0 and the fourth entry indicates w1 6= 0, which is a contradiction. Let fe1; e2; e3g be a basis for V1 and fu1; u2g be a basis for V2. Then a basis for V1 ⊗ V2 is fe1 ⊗ u1; e1 ⊗ u2; e2 ⊗ u1; e2 ⊗ u2; e3 ⊗ u1; e3 ⊗ u2; g 3 2 Exercise 2. Consider the claim mentioned above. Let V1 = R , V2 = R and let both ei and ui be standard basis vectors. Show that the basis fe1 ⊗ u1; e1 ⊗ u2; e2 ⊗ u1; e2 ⊗ u2; e3 ⊗ u1; e3 ⊗ u2; g 6 is the standard basis for the space R . Solution 2. Write out the Kronecker product for each one: • e1 ⊗ u1 = (1; 0; 0; 0; 0; 0) • e1 ⊗ u2 = (0; 1; 0; 0; 0; 0) • e2 ⊗ u1 = (0; 0; 1; 0; 0; 0) • e2 ⊗ u2 = (0; 0; 0; 1; 0; 0) • e3 ⊗ u1 = (0; 0; 0; 0; 1; 0) • e3 ⊗ u2 = (0; 0; 0; 0; 0; 1) Remark 0.1. Given two vector spaces over the same field, km; kn, their tensor product km ⊗ kn =∼ kmn. Similarly, km ⊗ kn =∼ kn ⊗ km. A lot of sloppiness in the treatment of tensors arises from this fact. Remark 0.2. In many resources, the preferred definition of a tensor product is one based on the “universal property.” A universal property is exactly what it sounds like–a property that must hold true regardless of the setting. With the rise of category theory in the 20th century, defining mathematical concepts using its universal property has become the norm, especially in geometry. Such definitions avoid a choice of representation (e.g. a choice of basis) and, as a result, are often heralded as more elegant. While I agree 1 with this view, I think such definitions can appear too mysterious to one just learning the subject. Despite the question of its helpfulness, I present the definition here: The tensor product of vector spaces V and W is a vector space V ⊗ W equipped with a fixed surjective bilinear map B : V × W ! V ⊗ W such that for any vector space S and bilinear map B^ : V × W ! S; there is a unique linear map T : V ⊗ W ! S such that B^ = T ◦ B. This is depicted as follows: B V × W V ⊗ W T B^ S Exercise 3. Use the universal property to show that if the tensor product exists, it must be unique. Solution 3. Let Y and Z both be tensor products of the vector spaces V; W . Then, we can observe the following two diagrams hold: B B^ V × W Y V × W Z T1 T2 B^ B Z Y This means B^ = T1 ◦ B, B = T2 ◦ B^, and, hence, B^ = T1 ◦ T2 ◦ B^ and B = T2 ◦ T1 ◦ B. Thus, T1 and T2 are invertible maps and must be isomorphisms. 0.1 Tensors as Maps. Let us now pay close attention to the subscripts of the vectors. Then the tensor product written previously (v1; v2; v3) ⊗ (w1; w2) = (v1w1; v1w2; v2w1; v2w2; v3w1; v3w2) could be reorganized as a matrix in the obvious way: 0 1 v1w1 v1w2 @ v2w1 v2w2 A v3w1 v3w2 2 3 Notice that this is a linear map whose domain is R and whose range is R . In this way, elements in 3 2 ∼ 2 3 R ⊗ R = L (R ; R ). 3 2 Exercise 4. Explain why this matrix must be rank 1. Does that mean all of R ⊗ R is congruent to only rank 1 matrices? 3 2 Solution 4. Observe that the columns are multiples of each other. Elements of R ⊗ R are all finite sums of rank 1 tensors, which means matrices of higher rank are constructible. Remark 0.3. “Rank” is a dangerous word in algebra, especially when talking about tensors. Above, we are considering the matrix rank. In physics, the rank is often used to denote the order of the tensor, like a vector is “rank 1,” a matrix is “rank 2,” etc. So when you search for “tensor rank” on the internet, you’ll likely find this definition. In algebraic geometry, the word “order” is used to denote this. In algebraic geometry, 2 the tensor rank, or simply the rank, is the minimal number of (matrix) rank 1 tensors needed to construct a tensor. If a tensor M can be constructed by 3 and no fewer elements of the form v ⊗ w, then M has (tensor) rank 3. Unfortunately, the terminology gets even worse with the introduction of the generic rank, sometimes denoted as “grank.” We may discuss this later in the course. There is a very nifty theorem from linear algebra called the Riesz Representation Theorem. It states that 1 one can identify any ~x (from a finite dimensional space ) with a map '~x(·): V ! k via the inner product: '~x(·) = h~x; ·i : Moreover, every linear functional is of this form. In this sense, we can identify the linear operator repre- sented by 0 1 v1w1 v1w2 @ v2w1 v2w2 A v3w1 v3w2 with the map 0 1 v1 @ v2 A hw; ·i v3 Exercise 5. Let v = (2; 1; 3) and w = (−1; 10): Pick two vectors and demonstrate the above claim over R. Solution 5. The inner product over R is the dot product. If v and w are as above, then the matrix is 0 −2 20 1 @ −1 10 A : −3 30 We pick our two vectors to be the standard basis vectors. Then 0 −2 20 1 0 −2 1 0 2 1 1 −1 10 = −1 = 1 (w · e ) @ A 0 @ A @ A 1 −3 30 −3 3 and 0 −2 20 1 0 20 1 0 2 1 0 −1 10 = 10 = 1 (w · e ) @ A 1 @ A @ A 2 −3 30 30 3 ∼ ∼ ∗ This observation is just a rephrasing of V1 ⊗ V2 = L (V2;V1): In particular, it is that V1 ⊗ V2 = V1 ⊗ V2 . ∼ ∗ Exercise 6. Prove that when both V1 and V2 are finite dimensional, V1 ⊗ V2 = V1 ⊗ V2. ∼ Solution 6. It is clear that V1 ⊗ V2 = V2 ⊗ V1 since we can take each basis element ej ⊗ ui of V1 ⊗ V2 and ∼ ∗ map it to the basis element ui ⊗ ej of V2 ⊗ V1. By the above discussion V2 ⊗ V1 = V2 ⊗ V1 . If we employ ∗ ∼ ∗ the same logic with the bases, we get V1 ⊗ V2 = V2 ⊗ V1 . 1This theorem applies to infinite dimensions as well; however, V =∼ V ∗ is guaranteed to hold when V is finite dimensional. Given the scope of this class, we restrict to this case. 3 0.2 Composing Tensors. Tensors can be seen as vectors, as multi-dimensional arrays (e.g. matrices), and as linear maps. Let us focus in for a moment on how we can compose such maps. Let W be a vector space of dimension n, let fβ1; : : : βng be an orthonormal basis of W , and let f'β1 ;:::;'βn g be a corresponding basis of W ∗ such that ( 1 i = j 'β (βj) = : i 0 i 6= j 3 ∗ Exercise 7. Let W = R and let wi = ei, the standard basis. Find 'wi . Show that it is a basis for W . Solution 7. Let w = (w1; w2; w3). Then 'e1 (w) = e1 · w = w1: Similar results hold for e2 and e3. Consider the linear functional f. By the Riesz representation theorem, there exists a vector x 2 W such that f(·) = hx; ·i = ha1e1 + a2e2 + a3e3; ·i = a1 he1; ·i + a2 he2; ·i + a3 he3; ·i = a1'β1 (·) + a2'β2 (·) + a3'β3 (·) Based on the previous discussion, we can take any element in the space V ⊗ U ⊗ W and think of it as a map T : W ! V ⊗ U n X T (·) = ai(vi ⊗ ui)'wi (·) i Similarly, we can think of another element of W ⊗ X ⊗ Y as a map S : X ⊗ Y ! W n X S(·) = biwi'xi⊗yi (·) i Hence, using the maps S and T , we can construct a map R = T ◦ S that can be identified as an element of V ⊗ U ⊗ X ⊗ Y : n n X X T ◦ S = ai(vi ⊗ ui)'wi (·) ◦ biwi'xi⊗yi (·) i i n n X X = aibj(vi ⊗ ui)'wi (wj)'xj ⊗yj (·) i=1 j=1 n n X X = cij(vi ⊗ ui)'xj ⊗yj (·) i=1 j=1 Alternatively, we can write these two elements as matrices in (V ⊗ U) ⊗ W and W ⊗ (X ⊗ Y ) by treating (V ⊗ U) and (X ⊗ Y ) as their corresponding vector spaces.
Load more

Recommended publications
  • Block Kronecker Products and the Vecb Operator* CORE View
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Block Kronecker Products and the vecb Operator* Ruud H. Koning Department of Economics University of Groningen P.O. Box 800 9700 AV, Groningen, The Netherlands Heinz Neudecker+ Department of Actuarial Sciences and Econometrics University of Amsterdam Jodenbreestraat 23 1011 NH, Amsterdam, The Netherlands and Tom Wansbeek Department of Economics University of Groningen P.O. Box 800 9700 AV, Groningen, The Netherlands Submitted hv Richard Rrualdi ABSTRACT This paper is concerned with two generalizations of the Kronecker product and two related generalizations of the vet operator. It is demonstrated that they pairwise match two different kinds of matrix partition, viz. the balanced and unbalanced ones. Relevant properties are supplied and proved. A related concept, the so-called tilde transform of a balanced block matrix, is also studied. The results are illustrated with various statistical applications of the five concepts studied. *Comments of an anonymous referee are gratefully acknowledged. ‘This work was started while the second author was at the Indian Statistiral Institute, New Delhi. LINEAR ALGEBRA AND ITS APPLICATIONS 149:165-184 (1991) 165 0 Elsevier Science Publishing Co., Inc., 1991 655 Avenue of the Americas, New York, NY 10010 0024-3795/91/$3.50 166 R. H. KONING, H. NEUDECKER, AND T. WANSBEEK INTRODUCTION Almost twenty years ago Singh [7] and Tracy and Singh [9] introduced a generalization of the Kronecker product A@B. They used varying notation for this new product, viz. A @ B and A &3B. Recently, Hyland and Collins [l] studied the-same product under rather restrictive order conditions.
    [Show full text]
  • The Kronecker Product a Product of the Times
    The Kronecker Product A Product of the Times Charles Van Loan Department of Computer Science Cornell University Presented at the SIAM Conference on Applied Linear Algebra, Monterey, Califirnia, October 26, 2009 The Kronecker Product B C is a block matrix whose ij-th block is b C. ⊗ ij E.g., b b b11C b12C 11 12 C = b b ⊗ 21 22 b21C b22C Also called the “Direct Product” or the “Tensor Product” Every bijckl Shows Up c11 c12 c13 b11 b12 c21 c22 c23 b21 b22 ⊗ c31 c32 c33 = b11c11 b11c12 b11c13 b12c11 b12c12 b12c13 b11c21 b11c22 b11c23 b12c21 b12c22 b12c23 b c b c b c b c b c b c 11 31 11 32 11 33 12 31 12 32 12 33 b c b c b c b c b c b c 21 11 21 12 21 13 22 11 22 12 22 13 b21c21 b21c22 b21c23 b22c21 b22c22 b22c23 b21c31 b21c32 b21c33 b22c31 b22c32 b22c33 Basic Algebraic Properties (B C)T = BT CT ⊗ ⊗ (B C) 1 = B 1 C 1 ⊗ − − ⊗ − (B C)(D F ) = BD CF ⊗ ⊗ ⊗ B (C D) =(B C) D ⊗ ⊗ ⊗ ⊗ C B = (Perfect Shuffle)T (B C)(Perfect Shuffle) ⊗ ⊗ R.J. Horn and C.R. Johnson(1991). Topics in Matrix Analysis, Cambridge University Press, NY. Reshaping KP Computations 2 Suppose B, C IRn n and x IRn . ∈ × ∈ The operation y =(B C)x is O(n4): ⊗ y = kron(B,C)*x The equivalent, reshaped operation Y = CXBT is O(n3): y = reshape(C*reshape(x,n,n)*B’,n,n) H.V.
    [Show full text]
  • The Partition Algebra and the Kronecker Product (Extended Abstract)
    FPSAC 2013, Paris, France DMTCS proc. (subm.), by the authors, 1–12 The partition algebra and the Kronecker product (Extended abstract) C. Bowman1yand M. De Visscher2 and R. Orellana3z 1Institut de Mathematiques´ de Jussieu, 175 rue du chevaleret, 75013, Paris 2Centre for Mathematical Science, City University London, Northampton Square, London, EC1V 0HB, England. 3 Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755, USA Abstract. We propose a new approach to study the Kronecker coefficients by using the Schur–Weyl duality between the symmetric group and the partition algebra. Resum´ e.´ Nous proposons une nouvelle approche pour l’etude´ des coefficients´ de Kronecker via la dualite´ entre le groupe symetrique´ et l’algebre` des partitions. Keywords: Kronecker coefficients, tensor product, partition algebra, representations of the symmetric group 1 Introduction A fundamental problem in the representation theory of the symmetric group is to describe the coeffi- cients in the decomposition of the tensor product of two Specht modules. These coefficients are known in the literature as the Kronecker coefficients. Finding a formula or combinatorial interpretation for these coefficients has been described by Richard Stanley as ‘one of the main problems in the combinatorial rep- resentation theory of the symmetric group’. This question has received the attention of Littlewood [Lit58], James [JK81, Chapter 2.9], Lascoux [Las80], Thibon [Thi91], Garsia and Remmel [GR85], Kleshchev and Bessenrodt [BK99] amongst others and yet a combinatorial solution has remained beyond reach for over a hundred years. Murnaghan discovered an amazing limiting phenomenon satisfied by the Kronecker coefficients; as we increase the length of the first row of the indexing partitions the sequence of Kronecker coefficients obtained stabilises.
    [Show full text]
  • Kronecker Products
    Copyright ©2005 by the Society for Industrial and Applied Mathematics This electronic version is for personal use and may not be duplicated or distributed. Chapter 13 Kronecker Products 13.1 Definition and Examples Definition 13.1. Let A ∈ Rm×n, B ∈ Rp×q . Then the Kronecker product (or tensor product) of A and B is defined as the matrix a11B ··· a1nB ⊗ = . ∈ Rmp×nq A B . .. . (13.1) am1B ··· amnB Obviously, the same definition holds if A and B are complex-valued matrices. We restrict our attention in this chapter primarily to real-valued matrices, pointing out the extension to the complex case only where it is not obvious. Example 13.2. = 123 = 21 1. Let A 321and B 23. Then 214263 B 2B 3B 234669 A ⊗ B = = . 3B 2BB 634221 694623 Note that B ⊗ A = A ⊗ B. ∈ Rp×q ⊗ = B 0 2. For any B , I2 B 0 B . Replacing I2 by In yields a block diagonal matrix with n copies of B along the diagonal. 3. Let B be an arbitrary 2 × 2 matrix. Then b11 0 b12 0 0 b11 0 b12 B ⊗ I2 = . b21 0 b22 0 0 b21 0 b22 139 “ajlbook” — 2004/11/9 — 13:36 — page 139 — #147 From "Matrix Analysis for Scientists and Engineers" Alan J. Laub. Buy this book from SIAM at www.ec-securehost.com/SIAM/ot91.html Copyright ©2005 by the Society for Industrial and Applied Mathematics This electronic version is for personal use and may not be duplicated or distributed. 140 Chapter 13. Kronecker Products The extension to arbitrary B and In is obvious.
    [Show full text]
  • A Constructive Arbitrary-Degree Kronecker Product Decomposition of Tensors
    A CONSTRUCTIVE ARBITRARY-DEGREE KRONECKER PRODUCT DECOMPOSITION OF TENSORS KIM BATSELIER AND NGAI WONG∗ Abstract. We propose the tensor Kronecker product singular value decomposition (TKPSVD) that decomposes a real k-way tensor A into a linear combination of tensor Kronecker products PR (d) (1) with an arbitrary number of d factors A = j=1 σj Aj ⊗ · · · ⊗ Aj . We generalize the matrix (i) Kronecker product to tensors such that each factor Aj in the TKPSVD is a k-way tensor. The algorithm relies on reshaping and permuting the original tensor into a d-way tensor, after which a polyadic decomposition with orthogonal rank-1 terms is computed. We prove that for many (1) (d) different structured tensors, the Kronecker product factors Aj ;:::; Aj are guaranteed to inherit this structure. In addition, we introduce the new notion of general symmetric tensors, which includes many different structures such as symmetric, persymmetric, centrosymmetric, Toeplitz and Hankel tensors. Key words. Kronecker product, structured tensors, tensor decomposition, TTr1SVD, general- ized symmetric tensors, Toeplitz tensor, Hankel tensor AMS subject classifications. 15A69, 15B05, 15A18, 15A23, 15B57 1. Introduction. Consider the singular value decomposition (SVD) of the fol- lowing 16 × 9 matrix A~ 0 1:108 −0:267 −1:192 −0:267 −1:192 −1:281 −1:192 −1:281 1:102 1 B 0:417 −1:487 −0:004 −1:487 −0:004 −1:418 −0:004 −1:418 −0:228C B C B−0:127 1:100 −1:461 1:100 −1:461 0:729 −1:461 0:729 0:940 C B C B−0:748 −0:243 0:387 −0:243 0:387 −1:241 0:387 −1:241 −1:853C B C B 0:417
    [Show full text]
  • Introduction to Kronecker Products
    Math 515 Fall, 2010 Introduction to Kronecker Products If A is an m × n matrix and B is a p × q matrix, then the Kronecker product of A and B is the mp × nq matrix 2 3 a11B a12B ··· a1nB 6 7 6 a21B a22B ··· a2nB 7 A ⊗ B = 6 . 7 6 . 7 4 . 5 am1B am2B ··· amnB Note that if A and B are large matrices, then the Kronecker product A⊗B will be huge. MATLAB has a built-in function kron that can be used as K = kron(A, B); However, you will quickly run out of memory if you try this for matrices that are 50 × 50 or larger. Fortunately we can exploit the block structure of Kronecker products to do many compu- tations involving A ⊗ B without actually forming the Kronecker product. Instead we need only do computations with A and B individually. To begin, we state some very simple properties of Kronecker products, which should not be difficult to verify: • (A ⊗ B)T = AT ⊗ BT • If A and B are square and nonsingular, then (A ⊗ B)−1 = A−1 ⊗ B−1 • (A ⊗ B)(C ⊗ D) = AC ⊗ BD The next property we want to consider involves the matrix-vector multiplication y = (A ⊗ B)x; where A 2 Rm×n and B 2 Rp×q. Thus A ⊗ B 2 Rmp×nq, x 2 Rnq, and y 2 Rmp. Our goal is to exploit the block structure of the Kronecker product matrix to compute y without explicitly forming (A ⊗ B). To see how this can be done, first partition the vectors x and y as 2 3 2 3 x1 y1 6 x 7 6 y 7 6 2 7 q 6 2 7 p x = 6 .
    [Show full text]
  • Kronecker Products and Matrix Calculus in System Theory
    772 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-25, NO. 9, ISEFTEMBER 1978 Kronecker Products and Matrix Calculus in System Theory JOHN W. BREWER I Absfrucr-The paper begins with a review of the algebras related to and are based on the developments of Section IV. Also, a Kronecker products. These algebras have several applications in system novel and interesting operator notation will ble introduced theory inclluding the analysis of stochastic steady state. The calculus of matrk valued functions of matrices is reviewed in the second part of the in Section VI. paper. This cakuhs is then used to develop an interesting new method for Concluding comments are placed in Section VII. the identification of parameters of linear time-invariant system models. II. NOTATION I. INTRODUCTION Matrices will be denoted by upper case boldface (e.g., HE ART of differentiation has many practical ap- A) and column matrices (vectors) will be denoted by T plications in system analysis. Differentiation is used lower case boldface (e.g., x). The kth row of a matrix such. to obtain necessaryand sufficient conditions for optimiza- as A will be denoted A,. and the kth colmnn will be tion in a.nalytical studies or it is used to obtain gradients denoted A.,. The ik element of A will be denoted ujk. and Hessians in numerical optimization studies. Sensitiv- The n x n unit matrix is denoted I,,. The q-dimensional ity analysis is yet another application area for. differentia- vector which is “I” in the kth and zero elsewhereis called tion formulas. In turn, sensitivity analysis is applied to the the unit vector and is denoted design of feedback and adaptive feedback control sys- tems.
    [Show full text]
  • Representing Words in a Geometric Algebra
    Representing Words in a Geometric Algebra Arjun Mani Princeton University [email protected] Abstract Word embeddings, or the mapping of words to vectors, demonstrate interesting properties such as the ability to solve analogies. However, these models fail to capture many key properties of words (e.g. hierarchy), and one might ask whether there are objects that can better capture the complexity associated with words. Geometric algebra is a branch of mathematics that is characterized by algebraic operations on objects having geometric meaning. In this system objects are linear combinations of subspaces rather than vectors, and certain operations have the significance of intersecting or unioning these subspaces. In this paper we introduce and motivate geometric algebra as a better representation for word embeddings. Next we describe how to implement the geometric product and interestingly show that neural networks can learn this product. We then introduce a model that represents words as objects in this algebra and benchmark it on large corpuses; our results show some promise on traditional word embedding tasks. Thus, we lay the groundwork for further investigation of geometric algebra in word embeddings. 1 Motivation Words are complex creatures; a single word has one (or multiple) semantic meanings, plays a particular syntactic role in a sentence, and has complex relationships with different words (simple examples of such relationships are synonyms and antonyms). The representations of words as vectors, and the ability of these models to capture some of this complexity, has been a seminal advance in the field of natural language processing. Such word embedding models (first introduced by Mikolov et al.
    [Show full text]
  • Spectral Properties of Sums of Certain Kronecker Products
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 431 (2009) 1691–1701 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Spectral properties of sums of certain Kronecker products ∗ Jun-e Feng a, James Lam b, , Yimin Wei c a School of Mathematics, Shandong University, Jinan 250100, People’s Republic of China b Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong c School of Mathematics, Fudan University, Shanghai 20043, People’s Republic of China ARTICLE INFO ABSTRACT Article history: We introduce two kinds of sums of Kronecker products, and their Received 22 September 2008 induced operators. We study the algebraic properties of these two Accepted 4 June 2009 kinds of matrices and their associated operators; the properties Available online 4 July 2009 include their eigenvalues, their eigenvectors, and the relationships Submitted by C.K. Li between their spectral radii or spectral abscissae. Furthermore, two projected matrices of these Kronecker products and their induced Keywords: operators are also studied. Eigenvalue © 2009 Elsevier Inc. All rights reserved. Eigenvector Spectrum Kronecker product Spectral abscissa Spectral radius 1. Introduction Kronecker product, also called direct product or tensor product, is an operation which owes its origin from group theory and has important applications in physics [6]. Techniques involving Kronecker products have been successfully applied in many fields of matrix theory because of its computational and notational advantages, see [1,2,5,7,8,12,13] and references therein.
    [Show full text]
  • Determinants in the Kronecker Product of Matrices: the Incidence Matrix of a Complete Graph
    DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH CHRISTOPHER R.H. HANUSA AND THOMAS ZASLAVSKY Abstract. We investigate the least common multiple of all sub- determinants, lcmd(A⊗ B), of a Kronecker product of matrices, of which one is an integral matrix A with two columns and the other is the incidence matrix of a complete graph with n vertices. We prove that this quantity is the least common multiple of lcmd(A) to the power n − 1 and certain binomial functions of the entries of A. 1. Introduction In a study of non-attacking placements of chess pieces, Chaiken, Hanusa, and Zaslavsky [1] were led to a quasipolynomial formula that depends in part on the least common multiple of the determinants of all square submatrices of a certain Kronecker product matrix, namely, the Kronecker product of an integral 2 ×2 matrix A with the incidence matrix of a complete graph. We give a compact expression for the least common multiple of the subdeterminants of this product matrix, generalized to A of order m × 2. 2. Background Kronecker product. For matrices A =(aij)m×k and B =(bij)n×l, the Kronecker product A⊗B is defined to be the mn×kl block matrix a11B ··· a1kB . . .. . am1B ··· amkB It is known (see [2], for example) that when A and B are square matri- ces of orders m and n, respectively, then det(A⊗B)=det(A)n det(B)m. 2000 Mathematics Subject Classification. 15A15, 05C50, 15A57. Key words and phrases. Kronecker product, determinant, least common multi- ple, incidence matrix of complete graph, matrix minor.
    [Show full text]
  • Matrix Differential Calculus with Applications to Simple, Hadamard, and Kronecker Products
    JOURNAL OF MATHEMATICAL PSYCHOLOGY 29, 414492 (1985) Matrix Differential Calculus with Applications to Simple, Hadamard, and Kronecker Products JAN R. MAGNUS London School of Economics AND H. NEUDECKER University of Amsterdam Several definitions are in use for the derivative of an mx p matrix function F(X) with respect to its n x q matrix argument X. We argue that only one of these definitions is a viable one, and that to study smooth maps from the space of n x q matrices to the space of m x p matrices it is often more convenient to study the map from nq-space to mp-space. Also, several procedures exist for a calculus of functions of matrices. It is argued that the procedure based on differentials is superior to other methods of differentiation, and leads inter alia to a satisfactory chain rule for matrix functions. c! 1985 Academic Press, Inc 1. INTRODUCTION In mathematical psychology (among other disciplines) matrix calculus has become an important tool. Several definitions are in use for the derivative of a matrix function F(X) with respect to its matrix argument X (see Dwyer & MacPhail, 1948; Dwyer, 1967; Neudecker, 1969; McDonald & Swaminathan, 1973; MacRae, 1974; Balestra, 1976; Nel, 1980; Rogers, 1980; Polasek, 1984). We shall see that only one of these definitions is appropriate. Also, several procedures exist for a calculus of functions of matrices. We shall argue that the procedure based on differentials is superior to other methods of differentiation. These, then, are the two purposes of this paper. We have been inspired to write this paper by Bentler and Lee’s (1978) note.
    [Show full text]
  • Hadamard, Khatri-Rao, Kronecker and Other Matrix Products
    INTERNATIONAL JOURNAL OF °c 2008 Institute for Scienti¯c INFORMATION AND SYSTEMS SCIENCES Computing and Information Volume 4, Number 1, Pages 160{177 HADAMARD, KHATRI-RAO, KRONECKER AND OTHER MATRIX PRODUCTS SHUANGZHE LIU AND GOTZÄ TRENKLER This paper is dedicated to Heinz Neudecker, with a®ection and appreciation Abstract. In this paper we present a brief overview on Hadamard, Khatri- Rao, Kronecker and several related non-simple matrix products and their prop- erties. We include practical applications, in di®erent areas, of some of the al- gebraic results. Key Words. Hadamard (or Schur) product, Khatri-Rao product, Tracy- Singh product, Kronecker product, vector cross product, positive de¯nite vari- ance matrix, multi-way model, linear matrix equation, signal processing. 1. Introduction Matrices and matrix operations are studied and applied in many areas such as engineering, natural and social sciences. Books written on these topics include Berman et al. (1989), Barnett (1990), LÄutkepohl (1996), Schott (1997), Magnus and Neudecker (1999), Hogben (2006), Schmidt and Trenkler (2006), and espe- cially Bernstein (2005) with a number of referred books classi¯ed as for di®erent areas in the preface. In this paper, we will in particular overview briefly results (with practical applications) involving several non-simple matrix products, as they play a very important role nowadays and deserve to receive such attention. The Hadamard (or Schur) and Kronecker products are studied and applied widely in matrix theory, statistics, system theory and other areas; see, e.g. Styan (1973), Neudecker and Liu (1995), Neudecker and Satorra (1995), Trenkler (1995), Rao and Rao (1998), Zhang (1999), Liu (2000a, 2000b) and Van Trees (2002).
    [Show full text]