DOCSLIB.ORG

Guessing Linear Recurrence Relations of Sequence Tuples and P-Recursive Sequences with Linear Algebra Jérémy Berthomieu, Jean-Charles Faugère

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Guessing Linear Recurrence Relations of Sequence Tuples and P-Recursive Sequences with Linear Algebra Jérémy Berthomieu, Jean-Charles Faugère Guessing Linear Recurrence Relations of Sequence Tuples and P-recursive Sequences with Linear Algebra Jérémy Berthomieu, Jean-Charles Faugère To cite this version: Jérémy Berthomieu, Jean-Charles Faugère. Guessing Linear Recurrence Relations of Sequence Tuples and P-recursive Sequences with Linear Algebra. 41st International Symposium on Symbolic and Algebraic Computation, Jul 2016, Waterloo, ON, Canada. pp.95-102, 10.1145/2930889.2930926. hal-01314266 HAL Id: hal-01314266 https://hal.inria.fr/hal-01314266 Submitted on 11 May 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Guessing Linear Recurrence Relations of Sequence Tuples and P-recursive Sequences with Linear Algebra Jérémy Berthomieu Jean-Charles Faugère [email protected] [email protected] Sorbonne Universités, UPMC Univ Paris 06, CNRS, INRIA, Laboratoire d’Informatique de Paris 6 (LIP6), Équipe PolSys, 4 place Jussieu, 75252 Paris Cedex 05, France ABSTRACT Thanks to equations (2a) and (2b), we can compute any Given several n-dimensional sequences, we first present an term of b starting with b0;0 = 1. This makes b P-recursive. algorithm for computing the Gr¨obner basis of their module Deciding if a sequence is P-recursive, as defined by Stan- of linear recurrence relations. ley [28] finds applications for instance in Combinatorics and in Computer algebra [27]. We refer the reader to the series n A P-recursive sequence (ui)i2N satisfies linear recurrence relations with polynomial coefficients in i, as defined by of papers on planar [10, 11, 12] and 3D-space walks [7, 8]. P i1 in The generating series ui ;:::;i x ··· xn of a Stanley in 1980. Calling directly the aforementioned algo- i1;:::;in2N 1 n 1 j n rithm on the tuple of sequences (i u ) n for retrieving P-recursive sequence (ui1;:::;in )(i1;:::;in)2N is a D-finite func- i i2N j the relations yields redundant relations. Since the module of tion in x1; : : : ; xn, see [19, Theorem 3.7]. A D-finite function relations of a P-recursive sequence also has an extra struc- is such that the dimension of the vector space spanned by ture of a 0-dimensional right ideal of an Ore algebra, we all its derivatives is finite over K(x1; : : : ; xn), this allows one design a more efficient algorithm that takes advantage of to characterize such functions with initial conditions and this extra structure for computing the relations. differential equations. For instance, the Dynamical Dic- [3] generates dy- Finally, we show how to incorporate Gr¨obner bases com- tionary of Mathematical Functions namically and automatically web pages on elementary and putations in an Ore algebra K ht1; : : : ; tn; x1; : : : ; xni, with commutators x x − x x = t t − t t = t x − x t = 0 special functions thanks to the equations they satisfy. Like- k ` ` k k ` ` k k ` ` k wise, a P-recursive sequence is uniquely defined by the linear for k 6= ` and tk xk −xk tk = xk, into the algorithm designed for P-recursive sequences. This allows us to compute faster recurrence relations with polynomial coefficients it satisfies and a finite number of initial terms. the Gr¨obner basis of the ideal spanned by the first relations, such as in 2D/3D-space walks examples. 1.1 Related work CCS Concepts For one-dimensional sequences, C-relations (linear recur- rence relations with constant coefficients) can be guessed •Comput. method. ! Symbolic calculus algorithms; with Berlekamp { Massey algorithm [4, 20], while for multi- Keywords dimensional sequences, they can be guessed with Berlekamp { Massey { Sakata algorithm [24, 26] or Scalar-FGLM [5, 6]. Gr¨obner bases, P-recursive sequences, FGLM algorithm All these algorithms only guess and check that the relations 1. INTRODUCTION are valid for a finite number of terms of the sequence. Then, usually, one need to prove that they are satisfied for all the Computing linear recurrence relations of multi-dimensional terms of the sequence. In this paper, a table shall denote sequences is a fundamental problem in Computer Science. a finite subset of terms of a sequence: it is one of the in- As this is the core of the paper, we start with an exam- i ! put parameter of the algorithms since one cannot handle an ple. The binomial sequence b = ( 1 =(i2!(i1 − i2)!)) 2 (i1;i2)2N infinite sequence. satisfies Pascal's rule, a linear recurrence relation: Likewise, for one-dimensional sequences, P-relations (lin- 2 8 (i1; i2) 2 N ; bi1+1;i2+1 = bi1;i2+1 + bi1;i2 : (1) ear recurrence relations with polynomial coefficients) can be guessed with Beckermann { Labahn algorithm [2] for com- It has constant coefficients. But this sequence also satisfies puting Hermite-Pad´eapproximants. relations with polynomial coefficients in (i1; i2): When computing parameterized sums of multivariate se- quences, or integrals of multivariate functions, in general, it 8 (i ; i ) 2 2; (i − i + 1) b = (i + 1) b ; (2a) 1 2 N 1 2 i1+1;i2 1 i1;i2 is hopeless to obtain a closed form. Indeed, even for elemen- (i2 + 1) bi1;i2+1 = (i1 − i2) bi1;i2 : (2b) tary univariate function we may not have closed forms when computing their integral on some domain. This motivated Zeilberger in 1990 [29] to introduce the so-called creative telescoping method. It is now one of the most efficient ways to do so and has been a tremendously active research topic. We refer the reader to the up-to-date nice survey on the state ISSAC ’16 July 19–22, 2016, Waterloo, ON, Canada of the art in creative telescoping in [13] and the references ACM ISBN . therein. DOI: http://dx.doi.org/10.1145/2930889.2930926 Although sequences can satisfy some linear recurrence re- lations with constant or polynomial coefficients, they still ity of the linear system solving [9]. We also investigate the may be neither C-recursive nor P-recursive. Given a sub- number of table queries performed by the algorithms. set S of steps in f0; ±1g2, one can define the sequence u = (u ) 3 of the number of planar walks of length n n;i;j (n;i;j)2N ending at (i; j) with steps in S. Determining if u is P- 2. DEFINITIONS AND NOTATION recursive has applications in lattice path enumeration [1, We recall the classical definitions of a C-recursive sequence 11, 12]. Likewise, now 3D-space walks are investigated. and of a P-recursive sequence. We shall also see how to deal In [7], the authors use several different methods to prove with C-relations between terms of multiple sequences and that somes sequences of walks are P-recursive among which: see that from a P-recursive sequence, we can create a tuple the algebraic kernel method, the reduction to smaller dimen- of sequences satisfying C-relations. sions and the guessing and proving method. In all the paper, we shall use standard notation with bolded letters corresponding to vectors and sequences: for ∗ 1.2 Contributions n 2 N , i = (i1; : : : ; in) and x = (x1; : : : ; xn). As usual, i i1 in We start by recalling the definitions of C-recursive (linear x denotes x1 ··· xn and jij = i1 + ··· + in. Finally, n u = (ui)i2N is an n-dimensional sequence over a field K. recursive with constant coefficients) sequences and define C- n i i recursive tuples of sequences. This framework allows us to For i 2 N , we let [x ]u = ui = [x ], if no ambiguity on u transform a P-recursive sequence into a tuple of sequences. shall arise. We extend linearly this [ ] operator to K[x]. C-relations are straightforwardly extended to n-dimensional In Section 3, we extends Scalar-FGLM [5, 6] for guessing n the module of C-relations satisfied by a tuple of tables up to sequences [23, Definition 21]: let S be a finite subset of N , n let α = (αs)s2S be nonzero, then u = (ui)i2N satisfies the a degree bound d. The algorithm returns a (d+1)-truncated n Gr¨obner basis of their module of relations. The main idea of linear relation defined by α if for all i 2 N , this algorithm is the same as Scalar-FGLM's: extracting hX i+si X αs x = αs ui+s = 0: (3) a full-rank submatrix of maximal rank. s2S s2S P-relations with coefficients of degree at most δ satisfied n Operator [ ] allows us to make a one-to-one correspondence by a sequence u = (ui1;:::;in )(i1;:::;in)2N are C-relations sat- j1 jn between C-relations satisfied by u and polynomials in [x]: isfied by the tuple of sequences (i ··· i u ) n , K 1 n i1;:::;in (i1;:::;in)2N relation P α u = 0 is associated to polynomial P α xs. with j ; : : : ; j ≥ 0 and j + ··· + j ≤ δ. Hence, a first idea s2S s s s2S s 1 n 1 n According to Sakata [24] and to Fitzpatrick and Nor- is to compute them with the extended Scalar-FGLM for instance. However, this strategy is not optimal because the ton [17], a C-recursive sequence is defined as follows.
Load more

Recommended publications
  • The Geometry of Syzygies
    The Geometry of Syzygies A second course in Commutative Algebra and Algebraic Geometry David Eisenbud University of California, Berkeley with the collaboration of Freddy Bonnin, Clement´ Caubel and Hel´ ene` Maugendre For a current version of this manuscript-in-progress, see www.msri.org/people/staff/de/ready.pdf Copyright David Eisenbud, 2002 ii Contents 0 Preface: Algebra and Geometry xi 0A What are syzygies? . xii 0B The Geometric Content of Syzygies . xiii 0C What does it mean to solve linear equations? . xiv 0D Experiment and Computation . xvi 0E What’s In This Book? . xvii 0F Prerequisites . xix 0G How did this book come about? . xix 0H Other Books . 1 0I Thanks . 1 0J Notation . 1 1 Free resolutions and Hilbert functions 3 1A Hilbert’s contributions . 3 1A.1 The generation of invariants . 3 1A.2 The study of syzygies . 5 1A.3 The Hilbert function becomes polynomial . 7 iii iv CONTENTS 1B Minimal free resolutions . 8 1B.1 Describing resolutions: Betti diagrams . 11 1B.2 Properties of the graded Betti numbers . 12 1B.3 The information in the Hilbert function . 13 1C Exercises . 14 2 First Examples of Free Resolutions 19 2A Monomial ideals and simplicial complexes . 19 2A.1 Syzygies of monomial ideals . 23 2A.2 Examples . 25 2A.3 Bounds on Betti numbers and proof of Hilbert’s Syzygy Theorem . 26 2B Geometry from syzygies: seven points in P3 .......... 29 2B.1 The Hilbert polynomial and function. 29 2B.2 . and other information in the resolution . 31 2C Exercises . 34 3 Points in P2 39 3A The ideal of a finite set of points .
    [Show full text]
  • LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
    LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students.
    [Show full text]
  • Geometric Engineering of (Framed) Bps States
    GEOMETRIC ENGINEERING OF (FRAMED) BPS STATES WU-YEN CHUANG1, DUILIU-EMANUEL DIACONESCU2, JAN MANSCHOT3;4, GREGORY W. MOORE5, YAN SOIBELMAN6 Abstract. BPS quivers for N = 2 SU(N) gauge theories are derived via geometric engineering from derived categories of toric Calabi-Yau threefolds. While the outcome is in agreement of previous low energy constructions, the geometric approach leads to several new results. An absence of walls conjecture is formulated for all values of N, relating the field theory BPS spectrum to large radius D-brane bound states. Supporting evidence is presented as explicit computations of BPS degeneracies in some examples. These computations also prove the existence of BPS states of arbitrarily high spin and infinitely many marginal stability walls at weak coupling. Moreover, framed quiver models for framed BPS states are naturally derived from this formalism, as well as a mathematical formulation of framed and unframed BPS degeneracies in terms of motivic and cohomological Donaldson-Thomas invariants. We verify the conjectured absence of BPS states with \exotic" SU(2)R quantum numbers using motivic DT invariants. This application is based in particular on a complete recursive algorithm which determines the unframed BPS spectrum at any point on the Coulomb branch in terms of noncommutative Donaldson- Thomas invariants for framed quiver representations. Contents 1. Introduction 2 1.1. A (short) summary for mathematicians 7 1.2. BPS categories and mirror symmetry 9 2. Geometric engineering, exceptional collections, and quivers 11 2.1. Exceptional collections and fractional branes 14 2.2. Orbifold quivers 18 2.3. Field theory limit A 19 2.4.
    [Show full text]
  • THE RESULTANT of TWO POLYNOMIALS Case of Two
    THE RESULTANT OF TWO POLYNOMIALS PIERRE-LOÏC MÉLIOT Abstract. We introduce the notion of resultant of two polynomials, and we explain its use for the computation of the intersection of two algebraic curves. Case of two polynomials in one variable. Consider an algebraically closed field k (say, k = C), and let P and Q be two polynomials in k[X]: r r−1 P (X) = arX + ar−1X + ··· + a1X + a0; s s−1 Q(X) = bsX + bs−1X + ··· + b1X + b0: We want a simple criterion to decide whether P and Q have a common root α. Note that if this is the case, then P (X) = (X − α) P1(X); Q(X) = (X − α) Q1(X) and P1Q − Q1P = 0. Therefore, there is a linear relation between the polynomials P (X);XP (X);:::;Xs−1P (X);Q(X);XQ(X);:::;Xr−1Q(X): Conversely, such a relation yields a common multiple P1Q = Q1P of P and Q with degree strictly smaller than deg P + deg Q, so P and Q are not coprime and they have a common root. If one writes in the basis 1; X; : : : ; Xr+s−1 the coefficients of the non-independent family of polynomials, then the existence of a linear relation is equivalent to the vanishing of the following determinant of size (r + s) × (r + s): a a ··· a r r−1 0 ar ar−1 ··· a0 .. .. .. a a ··· a r r−1 0 Res(P; Q) = ; bs bs−1 ··· b0 b b ··· b s s−1 0 . .. .. .. bs bs−1 ··· b0 with s lines with coefficients ai and r lines with coefficients bj.
    [Show full text]
  • MATH 210A, FALL 2017 Question 1. Consider a Short Exact Sequence 0
    MATH 210A, FALL 2017 HW 3 SOLUTIONS WRITTEN BY DAN DORE, EDITS BY PROF.CHURCH (If you find any errors, please email [email protected]) α β Question 1. Consider a short exact sequence 0 ! A −! B −! C ! 0. Prove that the following are equivalent. (A) There exists a homomorphism σ : C ! B such that β ◦ σ = idC . (B) There exists a homomorphism τ : B ! A such that τ ◦ α = idA. (C) There exists an isomorphism ': B ! A ⊕ C under which α corresponds to the inclusion A,! A ⊕ C and β corresponds to the projection A ⊕ C C. α β When these equivalent conditions hold, we say the short exact sequence 0 ! A −! B −! C ! 0 splits. We can also equivalently say “β : B ! C splits” (since by (i) this only depends on β) or “α: A ! B splits” (by (ii)). Solution. (A) =) (B): Let σ : C ! B be such that β ◦ σ = idC . Then we can define a homomorphism P : B ! B by P = idB −σ ◦ β. We should think of this as a projection onto A, in that P maps B into the submodule A and it is the identity on A (which is exactly what we’re trying to prove). Equivalently, we must show P ◦ P = P and im(P ) = A. It’s often useful to recognize the fact that a submodule A ⊆ B is a direct summand iff there is such a projection. Now, let’s prove that P is a projection. We have β ◦ P = β − β ◦ σ ◦ β = β − idC ◦β = 0. Thus, by the universal property of the kernel (as developed in HW2), P factors through the kernel α: A ! B.
    [Show full text]
  • Basic Vocabulary 2 Generators and Relations for an R-Module
    Modules as given by matrices November 23 Material covered in Artin Ed 1: Chapter 12 section 1-6; and Artin Ed 2: 14.1-14.8; and also more generally covered in Dummit and Foote Abstract Algebra. 1 Recall modules; set up conventions; basic vocabulary We will work with commutative rings with unit R. The category of R-modules. If U; V are R- modules we have the direct sum U ⊕ V as an R-module, and HomR(U; V ). Some basic vocabulary, some of which we have introduced, and some of which we will be introducing shortly. • submodule, quotient module • finitely generated • cyclic R-modules and Ideals in R • free • \short" exact sequence of R-modules 0 ! U ! V ! W ! 0: • split \short" exact sequence of R-modules • vector spaces; dimensions, ranks. 2 Generators and relations for an R-module; presentations Recall that our rings are commutative with unit, and we're talking about modules over such rings. Recall what it means for a set of elements fx1; x2; : : : ; xng ⊂ M to be a \a system of generators" for M over R: it means that any element of M is a linear combo of these elements with coefficients in R. Definition 1 If M is an R-module and fx1; x2; : : : ; xng ⊂ M a subset of elements that has the property that any linear relation of the form n Σi=1rixi = 0 in the ring R is trivial in the sense that all the ri's are 0, we'll say that the elements fx1; x2; : : : ; xng are linearly independent over R.
    [Show full text]
  • Geometry of Syzygies Edward Pearce Mathematics Institute
    Geometry of Syzygies by Edward Pearce MA4K9 Research Project Submitted to The University of Warwick Mathematics Institute April, 2016 Geometry of Syzygies Edward Pearce 1203384 30th May 2016 Contents 1 Introduction 2 2 Free Resolutions 2 2.1 Graded Rings . .2 2.2 Construction of a free resolution . .3 2.3 Application - The Hilbert Function and Polynomial . .4 2.4 Minimal Free Resolutions and Hilbert Syzygy Theorem . .6 2.5 Graded Betti Numbers and Betti Diagrams . 11 3 Proof of Hilbert's Syzygy Theorem 14 3.1 Monomial Ideals and Simplicial Complexes . 14 3.2 Syzygies of Monomial Ideals . 18 3.3 Taylor and Koszul complexes . 21 3.4 Hilbert's Syzygy Theorem - a strengthening . 23 4 Castelnuovo-Mumford Regularity 25 4.1 Denition and rst applications . 25 4.2 The Regularity of a Cohen-Macaulay Module . 27 5 Toric Ideals 31 6 Eisenbud-Goto Conjecture 33 6.1 A General Regularity Conjecture . 33 6.2 Examples using Macaulay 2 . 34 A Additional Macaulay2 Code 36 1 1 Introduction In this project we introduce several important and useful concepts and techniques from commutative and homological algebra which can prove fruitful in the study of projective geometry through means of calculating numerical invariants and convey- ing other algebraic and geometric information about ideals and their corresponding varieties. There is also some interplay with combinatorics along the way. We build our toolkit in order to consider a conjectural connection which states that under particular assumptions on a projective variety we may bound the degree of the min- imal homogeneous generators of its associated ideal in terms of geometric data.
    [Show full text]
  • A Rough Guide to Linear Algebra
    A rough guide to linear algbera Dongryul Kim Version of April 18, 2020 ii Contents Preface v 1 Sets 1 1.1 Sets and maps . .1 1.2 Products, coproducts, and sets of maps . .4 1.3 Fun with diagrams . .7 1.4 Equivalence classes . .9 2 Vector spaces 13 2.1 Fields . 13 2.2 Vector spaces . 15 2.3 Matrices . 18 2.4 Products and direct sums . 21 2.5 Subspaces and quotients . 27 2.6 Vector spaces from linear maps . 29 2.7 Bases and dimension . 34 2.8 Dual spaces . 44 2.9 Linear algebra in combinatorics . 50 3 Multilinear algebra 55 3.1 Bilinear maps and tensor products . 55 3.2 Symmetric and exterior algebras . 62 3.3 The determinant . 68 3.4 Computing the inverse matrix . 72 4 Linear algebra without division 79 4.1 Commutative rings . 79 4.2 Modules . 82 4.3 Classification of finitely generated modules over a PID . 86 4.4 Frobenius and Jordan normal form . 97 4.5 Eigenvalues and eigenvectors . 103 iii iv CONTENTS 5 Linear algebra over R and C 109 5.1 A bit of analysis . 109 5.2 Inner products . 112 5.3 Operators on Hilbert spaces . 118 5.4 The spectral theorem . 121 5.5 Positivity of operators . 125 5.6 Duality in linear programming . 130 Epilogue 131 Preface At a certain point, one adopts a mode of learning mathematics. Then at a later point, after progressing through various styles of teaching and writing, one develops one's own point of view for how mathematics is to be taught.
    [Show full text]
  • 3. Modules 27
    3. Modules 27 3. Modules In linear algebra, the most important structure is that of a vector space over a field. For commutative algebra it is therefore useful to consider the generalization of this concept to the case where the underlying space of scalars is a commutative ring R instead of a field. The resulting structure is called a module; we will introduce and study it in this chapter. In fact, there is another more subtle reason why modules are very powerful: they unify many other structures that you already know. For example, when you first heard about quotient rings you were probably surprised that in order to obtain a quotient ring R=I one needs an ideal I of R, i. e. a structure somewhat different from that of a (sub-)ring. In contrast, we will see in Example 3.4 (a) that ideals as well as quotient rings of R are just special cases of modules over R, so that one can deal with both these structures in the same way. Even more unexpectedly, it turns out that modules over certain rings allow a special interpretation: modules over Z are nothing but Abelian groups, whereas a module over the polynomial ring K[x] over a field K is exactly the same as a K-vector space V together with a linear map j : V ! V (see Examples 3.2 (d) and 3.8, respectively). Consequently, general results on modules will have numerous consequences in many different setups. So let us now start with the definition of modules. In principle, their theory that we will then quickly discuss in this chapter is entirely analogous to that of vector spaces [G2, Chapters 13 to 18].
    [Show full text]
  • Jordan Normal Form
    Jordan normal form 1 Principal ideal domains A commutative ring R is an integral domain if it has no zero divisors, i.e., a · b = 0 for some a, b ∈ R implies a = 0 or b = 0. An integral domain R is a principal ideal domain (PID) if every ideal in R is generated by a single element. Examples of PID-s include Z and polynomial rings F [x] of a single variable over a field F . Both examples have a norm function N : R \{0} → N that satisfies the Euclidean property: for all a ∈ R and b ∈ R \{0} there is a q ∈ R and and r ∈ R such that a = qb + r and r = 0 or N(r) < N(b). (For R = Z, the norm function may be chosen to be the absolute value, for R = F [x] the norm function may be chosen as the degree function.) Such domains are called Euclidean domains and they have a Euclidean algorithm. Lemma 1 Every Euclidean domain R is a PID. Proof: Let I be a nonzero ideal of R and let b ∈ I \{0} be an element whose norm is smallest among the norms of elements of I \{0}. We claim that I = (b). In fact, for any a ∈ I, we may write a = qb+r such that r = 0 or N(r) < N(b). If r = 0 then a = qb ∈ (b). If r 6= 0 then r = a − qb ∈ I \{0} has smaller norm than b, in contradiction with the choice of b. Thus we must have a ∈ (b) for all a ∈ I, and so I ⊆ (b).
    [Show full text]
  • Linear Algebra
    AAA Part IB of the Mathematical Tripos of the University of Cambridge Michaelmas 2012 Linear Algebra Lectured by: Notes by: Prof. I. Grojnowski Alex Chan Comments and corrections should be sent to [email protected]. This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. The following resources are not endorsed by the University of Cambridge. Printed Friday, 11 January 2013. Course schedule Definition of a vector space (over R or C), subspaces, the space spanned by a subset. Linear independence, bases, dimension. Direct sums and complementary subspaces. [3] Linear maps, isomorphisms. Relation between rank and nullity. The space of linear maps from U to V , representation by matrices. Change of basis. Row rank and column rank. [4] Determinant and trace of a square matrix. Determinant of a product of two matrices and of the inverse matrix. Determinant of an endomorphism. The adjugate matrix. [3] Eigenvalues and eigenvectors. Diagonal and triangular forms. Characteristic and min- imal polynomials. Cayley-Hamilton Theorem over C. Algebraic and geometric multi- plicity of eigenvalues. Statement and illustration of Jordan normal form. [4] Dual of a finite-dimensional vector space, dual bases and maps. Matrix representation, rank and determinant of dual map. [2] Bilinear forms. Matrix representation, change of basis. Symmetric forms and their link with quadratic forms. Diagonalisation of quadratic forms. Law of inertia, classification by rank and signature. Complex Hermitian forms. [4] Inner product spaces, orthonormal sets, orthogonal projection, V = W ⊕ W ?. Gram- Schmidt orthogonalisation. Adjoints. Diagonalisation of Hermitian matrices. Orthogo- nality of eigenvectors and properties of eigenvalues.
    [Show full text]
  • Linear Algebra for Math 542
    Linear Algebra for Math 542 JWR Spring 2001 2 Contents 1 Preliminaries 7 1.1 Sets and Maps . 7 1.2 Matrix Theory . 11 2 Vector Spaces 13 2.1 Vector Spaces . 14 2.2 Linear Maps . 15 2.3 Space of Linear Maps . 18 2.4 Frames and Matrix Representation . 19 2.5 Null Space and Range . 22 2.6 Subspaces . 23 2.7 Examples . 24 2.7.1 Matrices . 24 2.7.2 Polynomials . 25 2.7.3 Trigonometric Polynomials . 27 2.7.4 Derivative and Integral . 30 2.8 Exercises . 31 3 Bases and Frames 37 3.1 Maps and Sequences . 37 3.2 Independence . 39 3.3 Span . 41 3.4 Basis and Frame . 42 3.5 Examples and Exercises . 43 3.6 Cardinality . 49 3.7 The Dimension Theorem . 50 3.8 Isomorphism . 51 3.9 Extraction . 52 3 4 CONTENTS 3.10 Extension . 54 3.11 One-sided Inverses . 55 3.12 Independence and Span . 57 3.13 Rank and Nullity . 57 3.14 Exercises . 58 4 Matrix Representation 63 4.1 The Representation Theorem . 63 4.2 The Transition Matrix . 67 4.3 Change of Frames . 69 4.4 Flags . 73 4.5 Normal Forms . 74 4.5.1 Zero-One Normal Form . 75 4.5.2 Row Echelon Form . 78 4.5.3 Reduced Row Echelon Form . 79 4.5.4 Diagonalization . 80 4.5.5 Triangular Matrices . 81 4.5.6 Strictly Triangular Matrices . 82 4.6 Exercises . 83 5 Block Diagonalization 89 5.1 Direct Sums . 89 5.2 Idempotents .
    [Show full text]