2 Span, Basis, and Rank 2.1 Linear Combinations
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ADDITIVE MAPS on RANK K BIVECTORS 1. Introduction. Let N 2 Be an Integer and Let F Be a Field. We Denote by M N(F) the Algeb
Electronic Journal of Linear Algebra, ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 36, pp. 847-856, December 2020. ADDITIVE MAPS ON RANK K BIVECTORS∗ WAI LEONG CHOOIy AND KIAM HEONG KWAy V2 Abstract. Let U and V be linear spaces over fields F and K, respectively, such that dim U = n > 2 and jFj > 3. Let U n−1 V2 V2 be the second exterior power of U. Fixing an even integer k satisfying 2 6 k 6 n, it is shown that a map : U! V satisfies (u + v) = (u) + (v) for all rank k bivectors u; v 2 V2 U if and only if is an additive map. Examples showing the indispensability of the assumption on k are given. Key words. Additive maps, Second exterior powers, Bivectors, Ranks, Alternate matrices. AMS subject classifications. 15A03, 15A04, 15A75, 15A86. 1. Introduction. Let n > 2 be an integer and let F be a field. We denote by Mn(F) the algebra of n×n matrices over F. Given a nonempty subset S of Mn(F), a map : Mn(F) ! Mn(F) is called commuting on S (respectively, additive on S) if (A)A = A (A) for all A 2 S (respectively, (A + B) = (A) + (B) for all A; B 2 S). In 2012, using Breˇsar'sresult [1, Theorem A], Franca [3] characterized commuting additive maps on invertible (respectively, singular) matrices of Mn(F). He continued to characterize commuting additive maps on rank k matrices of Mn(F) in [4], where 2 6 k 6 n is a fixed integer, and commuting additive maps on rank one matrices of Mn(F) in [5]. -
Introduction to Linear Bialgebra
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of New Mexico University of New Mexico UNM Digital Repository Mathematics and Statistics Faculty and Staff Publications Academic Department Resources 2005 INTRODUCTION TO LINEAR BIALGEBRA Florentin Smarandache University of New Mexico, [email protected] W.B. Vasantha Kandasamy K. Ilanthenral Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebra Commons, Analysis Commons, Discrete Mathematics and Combinatorics Commons, and the Other Mathematics Commons Recommended Citation Smarandache, Florentin; W.B. Vasantha Kandasamy; and K. Ilanthenral. "INTRODUCTION TO LINEAR BIALGEBRA." (2005). https://digitalrepository.unm.edu/math_fsp/232 This Book is brought to you for free and open access by the Academic Department Resources at UNM Digital Repository. It has been accepted for inclusion in Mathematics and Statistics Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. INTRODUCTION TO LINEAR BIALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-mail: [email protected] K. Ilanthenral Editor, Maths Tiger, Quarterly Journal Flat No.11, Mayura Park, 16, Kazhikundram Main Road, Tharamani, Chennai – 600 113, India e-mail: [email protected] HEXIS Phoenix, Arizona 2005 1 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. -
Do Killingâ•Fiyano Tensors Form a Lie Algebra?
University of Massachusetts Amherst ScholarWorks@UMass Amherst Physics Department Faculty Publication Series Physics 2007 Do Killing–Yano tensors form a Lie algebra? David Kastor University of Massachusetts - Amherst, [email protected] Sourya Ray University of Massachusetts - Amherst Jennie Traschen University of Massachusetts - Amherst, [email protected] Follow this and additional works at: https://scholarworks.umass.edu/physics_faculty_pubs Part of the Physics Commons Recommended Citation Kastor, David; Ray, Sourya; and Traschen, Jennie, "Do Killing–Yano tensors form a Lie algebra?" (2007). Classical and Quantum Gravity. 1235. Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/1235 This Article is brought to you for free and open access by the Physics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Physics Department Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. Do Killing-Yano tensors form a Lie algebra? David Kastor, Sourya Ray and Jennie Traschen Department of Physics University of Massachusetts Amherst, MA 01003 ABSTRACT Killing-Yano tensors are natural generalizations of Killing vec- tors. We investigate whether Killing-Yano tensors form a graded Lie algebra with respect to the Schouten-Nijenhuis bracket. We find that this proposition does not hold in general, but that it arXiv:0705.0535v1 [hep-th] 3 May 2007 does hold for constant curvature spacetimes. We also show that Minkowski and (anti)-deSitter spacetimes have the maximal num- ber of Killing-Yano tensors of each rank and that the algebras of these tensors under the SN bracket are relatively simple exten- sions of the Poincare and (A)dS symmetry algebras. -
21. Orthonormal Bases
21. Orthonormal Bases The canonical/standard basis 011 001 001 B C B C B C B0C B1C B0C e1 = B.C ; e2 = B.C ; : : : ; en = B.C B.C B.C B.C @.A @.A @.A 0 0 1 has many useful properties. • Each of the standard basis vectors has unit length: q p T jjeijj = ei ei = ei ei = 1: • The standard basis vectors are orthogonal (in other words, at right angles or perpendicular). T ei ej = ei ej = 0 when i 6= j This is summarized by ( 1 i = j eT e = δ = ; i j ij 0 i 6= j where δij is the Kronecker delta. Notice that the Kronecker delta gives the entries of the identity matrix. Given column vectors v and w, we have seen that the dot product v w is the same as the matrix multiplication vT w. This is the inner product on n T R . We can also form the outer product vw , which gives a square matrix. 1 The outer product on the standard basis vectors is interesting. Set T Π1 = e1e1 011 B C B0C = B.C 1 0 ::: 0 B.C @.A 0 01 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 0 . T Πn = enen 001 B C B0C = B.C 0 0 ::: 1 B.C @.A 1 00 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 1 In short, Πi is the diagonal square matrix with a 1 in the ith diagonal position and zeros everywhere else. -
Multivector Differentiation and Linear Algebra 0.5Cm 17Th Santaló
Multivector differentiation and Linear Algebra 17th Santalo´ Summer School 2016, Santander Joan Lasenby Signal Processing Group, Engineering Department, Cambridge, UK and Trinity College Cambridge [email protected], www-sigproc.eng.cam.ac.uk/ s jl 23 August 2016 1 / 78 Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. 2 / 78 Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. 3 / 78 Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. 4 / 78 Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... 5 / 78 Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary 6 / 78 We now want to generalise this idea to enable us to find the derivative of F(X), in the A ‘direction’ – where X is a general mixed grade multivector (so F(X) is a general multivector valued function of X). Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi. Then, provided A has same grades as X, it makes sense to define: F(X + tA) − F(X) A ∗ ¶XF(X) = lim t!0 t The Multivector Derivative Recall our definition of the directional derivative in the a direction F(x + ea) − F(x) a·r F(x) = lim e!0 e 7 / 78 Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi. -
Appendix a Spinors in Four Dimensions
Appendix A Spinors in Four Dimensions In this appendix we collect the conventions used for spinors in both Minkowski and Euclidean spaces. In Minkowski space the flat metric has the 0 1 2 3 form ηµν = diag(−1, 1, 1, 1), and the coordinates are labelled (x ,x , x , x ). The analytic continuation into Euclidean space is madethrough the replace- ment x0 = ix4 (and in momentum space, p0 = −ip4) the coordinates in this case being labelled (x1,x2, x3, x4). The Lorentz group in four dimensions, SO(3, 1), is not simply connected and therefore, strictly speaking, has no spinorial representations. To deal with these types of representations one must consider its double covering, the spin group Spin(3, 1), which is isomorphic to SL(2, C). The group SL(2, C) pos- sesses a natural complex two-dimensional representation. Let us denote this representation by S andlet us consider an element ψ ∈ S with components ψα =(ψ1,ψ2) relative to some basis. The action of an element M ∈ SL(2, C) is β (Mψ)α = Mα ψβ. (A.1) This is not the only action of SL(2, C) which one could choose. Instead of M we could have used its complex conjugate M, its inverse transpose (M T)−1,or its inverse adjoint (M †)−1. All of them satisfy the same group multiplication law. These choices would correspond to the complex conjugate representation S, the dual representation S,and the dual complex conjugate representation S. We will use the following conventions for elements of these representations: α α˙ ψα ∈ S, ψα˙ ∈ S, ψ ∈ S, ψ ∈ S. -
Determinants Linear Algebra MATH 2010
Determinants Linear Algebra MATH 2010 • Determinants can be used for a lot of different applications. We will look at a few of these later. First of all, however, let's talk about how to compute a determinant. • Notation: The determinant of a square matrix A is denoted by jAj or det(A). • Determinant of a 2x2 matrix: Let A be a 2x2 matrix a b A = c d Then the determinant of A is given by jAj = ad − bc Recall, that this is the same formula used in calculating the inverse of A: 1 d −b 1 d −b A−1 = = ad − bc −c a jAj −c a if and only if ad − bc = jAj 6= 0. We will later see that if the determinant of any square matrix A 6= 0, then A is invertible or nonsingular. • Terminology: For larger matrices, we need to use cofactor expansion to find the determinant of A. First of all, let's define a few terms: { Minor: A minor, Mij, of the element aij is the determinant of the matrix obtained by deleting the ith row and jth column. ∗ Example: Let 2 −3 4 2 3 A = 4 6 3 1 5 4 −7 −8 Then to find M11, look at element a11 = −3. Delete the entire column and row that corre- sponds to a11 = −3, see the image below. Then M11 is the determinant of the remaining matrix, i.e., 3 1 M11 = = −8(3) − (−7)(1) = −17: −7 −8 ∗ Example: Similarly, to find M22 can be found looking at the element a22 and deleting the same row and column where this element is found, i.e., deleting the second row, second column: Then −3 2 M22 = = −8(−3) − 4(−2) = 16: 4 −8 { Cofactor: The cofactor, Cij is given by i+j Cij = (−1) Mij: Basically, the cofactor is either Mij or −Mij where the sign depends on the location of the element in the matrix. -
Handout 9 More Matrix Properties; the Transpose
Handout 9 More matrix properties; the transpose Square matrix properties These properties only apply to a square matrix, i.e. n £ n. ² The leading diagonal is the diagonal line consisting of the entries a11, a22, a33, . ann. ² A diagonal matrix has zeros everywhere except the leading diagonal. ² The identity matrix I has zeros o® the leading diagonal, and 1 for each entry on the diagonal. It is a special case of a diagonal matrix, and A I = I A = A for any n £ n matrix A. ² An upper triangular matrix has all its non-zero entries on or above the leading diagonal. ² A lower triangular matrix has all its non-zero entries on or below the leading diagonal. ² A symmetric matrix has the same entries below and above the diagonal: aij = aji for any values of i and j between 1 and n. ² An antisymmetric or skew-symmetric matrix has the opposite entries below and above the diagonal: aij = ¡aji for any values of i and j between 1 and n. This automatically means the digaonal entries must all be zero. Transpose To transpose a matrix, we reect it across the line given by the leading diagonal a11, a22 etc. In general the result is a di®erent shape to the original matrix: a11 a21 a11 a12 a13 > > A = A = 0 a12 a22 1 [A ]ij = A : µ a21 a22 a23 ¶ ji a13 a23 @ A > ² If A is m £ n then A is n £ m. > ² The transpose of a symmetric matrix is itself: A = A (recalling that only square matrices can be symmetric). -
Geometric-Algebra Adaptive Filters Wilder B
1 Geometric-Algebra Adaptive Filters Wilder B. Lopes∗, Member, IEEE, Cassio G. Lopesy, Senior Member, IEEE Abstract—This paper presents a new class of adaptive filters, namely Geometric-Algebra Adaptive Filters (GAAFs). They are Faces generated by formulating the underlying minimization problem (a deterministic cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well- Edges suited for the description of geometric transformations. Also, (directed lines) differently from standard adaptive-filtering theory, Geometric Calculus (the extension of GA to differential calculus) allows Fig. 1. A polyhedron (3-dimensional polytope) can be completely described for applying the same derivation techniques regardless of the by the geometric multiplication of its edges (oriented lines, vectors), which type (subalgebra) of the data, i.e., real, complex numbers, generate the faces and hypersurfaces (in the case of a general n-dimensional quaternions, etc. Relying on those characteristics (among others), polytope). a deterministic quadratic cost function is posed, from which the GAAFs are devised, providing a generalization of regular adaptive filters to subalgebras of GA. From the obtained update rule, it is shown how to recover the following least-mean squares perform calculus with hypercomplex quantities, i.e., elements (LMS) adaptive filter variants: real-entries LMS, complex LMS, that generalize complex numbers for higher dimensions [2]– and quaternions LMS. Mean-square analysis and simulations in [10]. a system identification scenario are provided, showing very good agreement for different levels of measurement noise. GA-based AFs were first introduced in [11], [12], where they were successfully employed to estimate the geometric Index Terms—Adaptive filtering, geometric algebra, quater- transformation (rotation and translation) that aligns a pair of nions. -
Matrices and Tensors
APPENDIX MATRICES AND TENSORS A.1. INTRODUCTION AND RATIONALE The purpose of this appendix is to present the notation and most of the mathematical tech- niques that are used in the body of the text. The audience is assumed to have been through sev- eral years of college-level mathematics, which included the differential and integral calculus, differential equations, functions of several variables, partial derivatives, and an introduction to linear algebra. Matrices are reviewed briefly, and determinants, vectors, and tensors of order two are described. The application of this linear algebra to material that appears in under- graduate engineering courses on mechanics is illustrated by discussions of concepts like the area and mass moments of inertia, Mohr’s circles, and the vector cross and triple scalar prod- ucts. The notation, as far as possible, will be a matrix notation that is easily entered into exist- ing symbolic computational programs like Maple, Mathematica, Matlab, and Mathcad. The desire to represent the components of three-dimensional fourth-order tensors that appear in anisotropic elasticity as the components of six-dimensional second-order tensors and thus rep- resent these components in matrices of tensor components in six dimensions leads to the non- traditional part of this appendix. This is also one of the nontraditional aspects in the text of the book, but a minor one. This is described in §A.11, along with the rationale for this approach. A.2. DEFINITION OF SQUARE, COLUMN, AND ROW MATRICES An r-by-c matrix, M, is a rectangular array of numbers consisting of r rows and c columns: ¯MM.. -
28. Exterior Powers
28. Exterior powers 28.1 Desiderata 28.2 Definitions, uniqueness, existence 28.3 Some elementary facts 28.4 Exterior powers Vif of maps 28.5 Exterior powers of free modules 28.6 Determinants revisited 28.7 Minors of matrices 28.8 Uniqueness in the structure theorem 28.9 Cartan's lemma 28.10 Cayley-Hamilton Theorem 28.11 Worked examples While many of the arguments here have analogues for tensor products, it is worthwhile to repeat these arguments with the relevant variations, both for practice, and to be sensitive to the differences. 1. Desiderata Again, we review missing items in our development of linear algebra. We are missing a development of determinants of matrices whose entries may be in commutative rings, rather than fields. We would like an intrinsic definition of determinants of endomorphisms, rather than one that depends upon a choice of coordinates, even if we eventually prove that the determinant is independent of the coordinates. We anticipate that Artin's axiomatization of determinants of matrices should be mirrored in much of what we do here. We want a direct and natural proof of the Cayley-Hamilton theorem. Linear algebra over fields is insufficient, since the introduction of the indeterminate x in the definition of the characteristic polynomial takes us outside the class of vector spaces over fields. We want to give a conceptual proof for the uniqueness part of the structure theorem for finitely-generated modules over principal ideal domains. Multi-linear algebra over fields is surely insufficient for this. 417 418 Exterior powers 2. Definitions, uniqueness, existence Let R be a commutative ring with 1. -
Preserivng Pieces of Information in a Given Order in HRR and GA$ C$
Proceedings of the Federated Conference on ISBN 978-83-60810-22-4 Computer Science and Information Systems pp. 213–220 Preserivng pieces of information in a given order in HRR and GAc Agnieszka Patyk-Ło´nska Abstract—Geometric Analogues of Holographic Reduced Rep- Secondly, this method of encoding will not detect similarity resentations (GA HRR or GAc—the continuous version of between eye and yeye. discrete GA described in [16]) employ role-filler binding based A quantum-like attempt to tackle the problem of information on geometric products. Atomic objects are real-valued vectors in n-dimensional Euclidean space and complex statements belong to ordering was made in [1]—a version of semantic analysis, a hierarchy of multivectors. A property of GAc and HRR studied reformulated in terms of a Hilbert-space problem, is compared here is the ability to store pieces of information in a given order with structures known from quantum mechanics. In particular, by means of trajectory association. We describe results of an an LSA matrix representation [1], [10] is rewritten by the experiment: finding the alignment of items in a sequence without means of quantum notation. Geometric algebra has also been the precise knowledge of trajectory vectors. Index Terms—distributed representations, geometric algebra, used extensively in quantum mechanics ([2], [4], [3]) and so HRR, BSC, word order, trajectory associations, bag of words. there seems to be a natural connection between LSA and GAc, which is the ground for fututre work on the problem I. INTRODUCTION of preserving pieces of information in a given order. VER the years several attempts have been made to As far as convolutions are concerned, the most interesting O preserve the order in which the objects are to be remem- approach to remembering information in a given order has bered with the help of binding and superposition.