PIT problems in the light of and the noncommutative rank algorithm
|
|
- Augustus Norris
- 5 years ago
- Views:
Transcription
1 PIT problems in the light of and the noncommutative rank algorithm Gábor Ivanyos MTA SZTAKI Optimization, Complexity and Invariant Theory, IAS, June 4-8, 2018.
2 PIT problems in this talk Determinant: det(x 0 A 0 + x 1 A x k A k ) 0 exists a non-singular matrix in A = A 0,..., A 1 What is the rk A (commutative) rank of (= max rank in) A Constructive version (rank optimization): Find a matrix of max rank in A We assume square case most problems reducible to that
3 Overview Common block triangular forms of matrices Behavior of Wong sequences Module problems: from easy to hard If time left: Spaces spanned by unknown rank one matrices
4 Some notation M n (F ) = M n n (F ) Block matrices, holes in matrices: ( A B = C) A B 0 C Block (upper) triangular matrices: Matrix sets: ( {( } A A B = : B, Carbitrary ) C) ( A B C), A and C square
5 Notation (2) Product of sets: AU = {Au : A A, u U} (subspace when either A or U is a subspace) AB = {AB : A A, B B} (similarity): in the same orbit of conjugation by GL, changing the basis ( GL GL ): in the same orbit of (independent) left-right multiplication by GL changing the two bases independently
6 Oil and Vinegar signature schemes (Patarin (1997),...) Public key: P = (P 1,..., P k ) F [x] k x = (x 1,..., x n), deg P = 2 Message: a F k Valid signature: a solution of P(x) = a Private key (hidden structure): easy system: P s.t. P (y) = a P = P A, A GL n (F ) a linear change of variables easiness : P is linear in the first o variables: no terms x i x j with i, j {1,..., o} by a random substitution for x j (j = o + 1,..., n) we have a solvable linear system (with good chance) x 1,..., x o : oil variables ; x o+1,..., n vinegar variables
7 Oil and Vinegar (2) Key generation: choose such P randomly, and A randomly Tuning: choose the parameters k, o, n: P easy to solve hard to break Balanced O & V (Patarin 1997): n = 2o (and k o) Breaking Balanced O & V (Kipnis & Shamir 1998): P i = Q i + linear P i = Q i + linear, Q i = A T Q i A pick Q 0 = α i Q i random invertible with h.p. (for most P) Q 0 = A T Q 0 A Q 0 = α i Q i ) R i := Q 1 0 Q i R i = Q 1 0 Q i
8 Breaking Balanced O & V Q 0 = A T Q 0 A (Q 0 = α i Q i ) R i := Q0 1 Q i (R i = Q 1 0 Q i ) key property: R i = A 1 R i A easier than Q i = A T Q i A Proof. R i = A 1 Q 0 A T A T Q i A = A 1 R i ( ) ( ) ( ) A ( ) ( ) I Q i =, Q I i 1 I, I ( ) ( ) 2 ( ) ( ) R i = Q 0 1 I Q i = I
9 Breaking Balanced O & V (2) R i A 1 ( ) A unique common block triangular form of R i for most P up to lin. changes of the O and V variables separately do not disturb easiness find common block triangularization of R i O & V decomposition of Q i e.g. use the MeatAxe Kipnis & Shamir: simpler direct method (exploits specialties of the setting) Unbalanced O & V (Kipnis & Patarin 1999) better hardness : Bulygin, Petzoldt & Buchmann (2010)
10 Block triangular forms ( ) GAH, n t t zero lower left block reduces many problems to the diag. blocks e.g, finding full rk. A A; Find B A with invertible upper left block, B A with invertible lower block, λb + C will be invertible except for a few λs instability
11 Block triangular forms (2) The full (commutative) rank case: A 0 A invertible use A 0 as a bijection between the domain and range a prefect matchings: bipartite graphs digraphs New matrix space: A 1 0 A = {A 1 0 A : A A} I n, A0 1 A = A 1 0 AI GL GL A, inherits block triang. (GA 0 H) 1 GAH = H 1 A 1 0 G 1 GAH = H 1 A 1 0 AH natural action on A 1 0 A: conjugation X H 1 XH = two-sided action of GL GL preserving I
12 Block triangular forms (3) I n A, H 1 AH ( ), n t t zero block First t basis vectors span an H 1 AH-invariant subspace U U = H 1 U t-dim A-invariant subspace Remark: if I n A and dim AU dim U then AU = U. nontrivial strong components in digraphs Env(A) enveloping (matrix) algebra closure of A w.r.t. lin. comb. and multiplications transitive closure of digraphs A-invariant subspace: submodule for Env(A) (or for the free algebra)
13 Finding common invariant subspaces Quite well studied/understood Many of the methods: based on structure of Env(A) one-sided ideals, zero divisors for A = I, A 0 : factors of the minimum polynomials of A 0 general A: zero div. factoring min. pol. of good A Env(A) over algebraically closed fields: almost easy Depends on the computational model Representation size explosion? E.g. huge (composite) extensions 2 2 M 2n (Q) A... p n p n
14 Finding invariant subspaces - rationality issues over non-closed base fields (extensions not allowed) over finite fields: only randomized methods (in large char), factoring polynomials MeatAxe for group representations over Q only a partial decompositions Hardness of distinguishing full matrix algebras from division algebras over Q (Rónyai 1987): in some generalizations of the quaternions existence of zero divisors quadratic residuousity mod composite numbers finding zero divisors: factoring integers a motivation in conjecturing the regularity
15 Using blowups for block triangularization A = A M d (F ) (on F n F d ). Property A = (I n M d (F ))A = A (I n M d (F ) (this characterizes blowups) A U V = A (I n M d (F ))U (I n M d (F ))V I n M d (F )-invariant subspaces of F n F d : (I M d (F ))U = U U = U F d U = {u F n : u v U for some 0 v F d } Computing U: v 1,..., v d : basis for F d, u 1,..., u k: basis for U u i = u ij v j u ij F n U is spanned by u ij (i = 1,..., k j = 1,..., d)
16 Using blowups (2) Lower left zero blocks in blowups: A = A M d (F ) U, V (I M d (F ))-invariant subsp. U = U M d (F ), V = V M d (F ) A U V AU AV L.L.Z.B. in A L.L.Z.B. in A block triang forms of A block triang forms of A Application of constructive ncrank:find singular block triang of some blowup A block triang of A or an invertible element in some blowup A block triang A a block triang of A More serious applications in the next talk (?)
17 The Wong sequence Given A 0, A 1,..., A k M n (F ) A = A 0,..., A k rk A 0 = r < n, c = n r (co-rank of A 0 ) Idealistic goal: find case (1) A A s.t. rk A > r or case (2) U F n s.t. dim AU dim U c U 0 = (0), U j = A 1 0 U j, U j+1 = AU j (A 1 0 W : full inverse image of W at A 0) U 0 U 1... U l, U 0 U 1... U l..., U j,... stops inside im A 0 case (2) otherwise escapes from im A 0 : AU j im A 0 for some j length of the (escaping) Wong sequence l = min{j : AU j im A 0 }
18 Length 1 Wong sequence l = 1; basic case n = r + 1, A 0 = I r, r > 0 A ker A 0 im A 0 i: A i ker A 0 im A ( 0 B A i + λa 0 ) + λi b ( B ) + λi (b 0) b has rank > r if λ and is not an eigenvalue of B (F large enough) Blind algorithm compute rk (A i + λa 0 ) (i = 1,..., k, λ = λ 1,..., λ r+1 )
19 Length 1 - some examples Examples (long Wong sequences): A 0 = , k = 1, A 1 = i: rk (A 0 + A 1 ) > rk A 0 k = n > 1, A i = E ii : rk (A 0 + A i ) = rk A 0 Length one a nice property: independent of the basis for A preserved by GLn GL n preserved by base filed extension F = R; A 0, A i pos. szemidef. v ker A 0 \ ker A i = (im A 0 ) \ ker A i 0 v T A i v, but v T A 0 w = 0 for every w
20 Length 1 - examples (2) A i diagonal (i = 0,..., k) i, v: im A 0 ker A 0 = (0) ker A 0, A i -invariant (because A i A 0 = A 0 A i ) A i ker A 0 im A 0 ker A 0 ker A i Application: simplicity of finite extensions of Q: L: field extension of F = Q, L : F = n a L: F [a] = subring (=subfield) generated by F and a Task: find a s.t. L = F [a] Matrix representation of L a M a = matrix of x ax on L (n n) identify a with M a ; Facts: M a are simultaneously diagonalizable over C F [a] : F = #distinct eigenvalues of a
21 simplicity of extensions (2) a Ad Ma = matrix of X M a X XM a A := {Ad Ma : a L} n-dim subspace of M n 2(F ) = δ 1... δ n Ad diagonal in M n 2: Ad E ij = E ij E ij = (δ i δ j )E ij max. rank is n 2 n when δ i δ j for i j generalizes to direct sums of field extensions (over perfect base fields)
22 Short Wong sequences Key observation of Bläser, Jindal & Pandey (2017) A = A 0, A 1,..., A k A = A 0, A 1 (over F (x 1,..., x k )) A 1 = x 1A , x k A k A 0 not of max rank in A (F sufficiently large) A 0 not of max rank in A (rk = ncrk for pair A ) A 1 U l im A 0 U 1,..., U l Wong sequence for A 0 in A
23 Short Wong sequences (2) Assume basic case A 0 = ( Ir ), n = r + 1 A 1 U l = A 1l ker A 0 lower right entry of A 1 : nonzero degree l polynomial in x 1,..., x k has term a x i1... x il A 0 is not of max rank in A = A 0, x i1 A i x il A il A 0 is not of max rank in A = A 0, A i1,..., A il. Assume l length of Wong seq. for A. Then A 0 is of max rank in A 0, A 1,..., A k A 0 is of max rank in A 0, A i1,..., A il for every subset {i 1,..., i l } {1,..., k}
24 Short Wong sequences (3) Algorithm (Bläser, Jindal & Pandey (2017)) Input: A 0, A 1,..., A k and l k Output: A 0 A of rank > rk A 0 or: l IS TOO SMALL for every subset {i 1,..., i l } {1,..., k} try A 0 + l t=1 ω ta it for all (ω 1,..., ω l ) Ω l ( Ω = n) complexity (kn) l poly
25 Progress of Wong sequences Wong sequence U 0 = (0), U j = AA 1 0 A 0 U j im A 0 (j = 0,..., l 1) Lemma (BJP17 for case k = 2) Assume that rk A 0 = r < ncrk A. Then for every 1 j < l, dim U j dim U j 1 + ncrk A r. sufficient ( to ) prove for n = ncrk A ( basic case ) Ir A 0 = ; s = ncrk A, take an s s window of full ncrk containing the upper left r by r
26 Progress of Wong (2) ( ) Ir A 0 = F n = im A 0 ker A 0, block structure using (0) = U 0 < U 1 <... < U l 1 im A 0 B 1 B 12 B 17 B 18 B 19 B 21 B 2 B 27 B 28. A A = B 76 B 7 B 78 B 87 B 8 B 97 B 98 B jj square (I for A 0 ); B l+2,l 0 cyclically shift by n r (7 = l)
27 Progress of Wong (3) diagonal shifted by ncrk r to the right B 19 B 1 B 12 B 17 B 18 B 21 B 2 B 27 B 28. A B 76 B 7 B 78 B 87 B 8 B 97 B 98 B jj has ncrk A r columns by def of ncrk
28 Progress of Wong (3) A formal proof the lemma for 1 j < l: U j 1 = A 1 0 U j 1 = U j 1 ker A 0 for A 0 = ( Ir ) ; dim U j = dim AU j 1 dim U j 1 (full ncrank) = dim U j 1 + dim ker A 0 = dim U j 1 + n r
29 Approximating the commutative rank Bläser, Jindal & Pandey (2017) r = max. rk in A = A 1,..., A k goal: find A A: rk A (1 ɛ)r) l = min(k, 1/ɛ ) Iteration if rk A 0 (1 ɛr) then: length of Wong seq. for A 0 and x 1 A x k A k rk A 0 /(r rk A 0 ) (1 ɛ)/ɛ = 1/ɛ 1 try A 0 = A 0 + ω 1 A i ω l A il replace A 0 with A 0 if better terminate if no improvement Cost: (kn) 1/ɛ poly
30 Thin Wong sequences dim U j+1 = dim U j + 1 (j = 0,..., l 1) basic case n = rk A b 11 b 12 b 17 B 18 b 19 b 19 b 11 b 12 b 17 B 18 b 21 b 22 b 27 B 28 b 21 b 22 b 27 B b 76 b 77 B 78 b 76 b 77 B 78 B 87 B 88 b 97 B 98 b 97 B 98 B 87 B 88 (7 = l + 1) find A A with no zero diag entry in the big upper left part find A + λa 0 with invertible lower right diag block (B 88 )
31 Thin Wong sequences (2) hardness of rank of diagonal A i over F of constant size q 3: reduction from coloring with q colors: vertices: v 1,..., v k, edges e 1,..., e n +1 if e t = {v i, v j }, j > i (A i ) tt = 1 if e t = {v i, v j }, j < i (i = 1,..., k) 0 otherwise. special instances: pencils: A = A 0, A 1 A 1,..., A k of rank one: find smallest l, A il... A i1 ker A 0 im A 0 A A 0, A i1,..., A il poly method for A spanned by A 0 and unknown rank one matrices
32 Wong sequences - remarks & problems Triangularizable spaces of full rank:... A GL GL.... Would be length 1 for rk A n rank of the diagonal part triangularization by conjugation Dual Wong sequence could recover part of triang structure Shortening length of Wong with A 0 + λa i? Example: A 0 triangular, A 1,..., A k diagonal Nicer classes? Nice classes for length 2?
33 Wong sequences - remarks & problems (2) Length and blowup size length 2 current blowup size (sufficient to increase rk A 0 ) thinness at a single step block triang current blowup size rk A 0 /4 relation with final blowup size? commutative rank for bounded blowup size? Rank of generators rank one: blowup size 1 rank 2: current blowup size 2 (looks so) final blowup size???? in special cases, e.g., (skew) symmetric? rank c: bound on current blowup size?
34 Modules modules for the free algebra B = F X 1,..., X t n-dimensional (left) B-module: V = F n, : B V V bilinear commutes with of B: (a b) v = a (b v) Notation: av = a v input data: linear maps L 1,..., L t : V V (n n matrices) action of X 1,..., X t multiplication tables in groups could take smaller (finite dim.) B Isomorphisms V, V, given by L 1,..., L t M n (F ), L 1,..., L t M n (F ) φ : V V bijective linear X i φ(v) = φ(x i v) L i φ = φ L i
35 Module morphisms (2) Homomorphisms V, V, given by L 1,..., L t M n (F ), L 1,..., L t M n (F ) Hom(V, V ) := {φ : V V lin. : φ L i = L i φ} subspace of Lin n n(f ) solutions of the lin. constraints φ L i = L i φ Isomorphism: n = n, invertible transf. from Hom(V, V ) Isomorphism: a full rank matrix Hom(V, V ) (n = n) In P: Chistov, I & Karpinski (1997) over many fields; Brooksbank & Luks (08); I, Karpinski & Saxena (010) all fields Length 1 Wong sequence in a special case...
36 Module morphisms (3) submodule Common invariant subspaces for L i Block triangular form of L i simple (irreducible) modules: no proper submodules direct sum ( ) external : space V V Li ; action L i internal : V = V 1 V 2 (of subspaces), V 1, V 2 submodule block diagonal form for L i indecomposable modules: no such decomp. Krull-Schmidt: uniqueness of decomposition into indecomposables: isomorphism types and multiplicities factorization of numbers
37 Module isomorphism - the decision version An new ncrank-based method Key observation: Hom(V, V ) M d (F ) = Hom(V d, V d ) Consequence: (assume dim V = dim V = n): Proof. V = V ncrk (Hom(V, V ) = n V = V rk Hom(V, V ) = n ncrk Hom(V, V ) = n V d = V d for some d V = V by (Krull-Schmidt)
38 Hardness of injectivity Are spaces Hom(V, V ) special? NO: every matrix space is essentially Hom(V, V ) Construction: I, Karpinski & Saxena (2010) A 1,..., A k arbitrary n n V, V modules for F x 1,..., x n, x n+1 dim V = n + 1, dim V = n + k ( L j = ( L j = ) e j n + 1 n + 1 e j = 1 j (j n) ) Â j, n + k n + k ( ) Â j = A (j) 1 A (j) : jth columns of A k 1,..., A k (j n) Another slicing of the 3-tensor A: (Âj ) il = (A l ) ij ( ( ) In n L n+1 = L 0), n+1 = In n 0k k
39 Hardness of injectivity (2) ( B Hom(V, V ): n + k n + 1-matrices C = α 1 B M n(f ), α =. F n s.t.: α k CL j = L jc (j = 1,..., n) α ) Proof. B = A α, where A α = α 1 A α k A k CL j = Hom(V, V ) C α = ( B (j) ) L j C = ( α1 A (j) α k A (j) k )) = ( Aα ( ) (j) A α ) injective A α α nonsingular
40 Module isomorphism - the semisimple case V = V semisimple (the indecomposable components are simple) important property: every submodule is a direct summand Assume φ : Hom(V, V ) not invertible. Let V 0 ker φ simple, let V = V 0 W 0. Let V = V 1 V t, V i irreducible By Krull-Schmidt, i s.t. V i im φ and V i = V 0. ψ 0 : V 0 V i isomorphism extend to V V i V : ψ(v + w) := ψ 0 (v) (v V 0, w W ) ψ ker φ = V 0 im φ length 1 Wong
41 Semisimple module algorithm - remarks Actually, finds max rank morphisms between semisimple module An application to decrease dimension of representation of simple algebras (Babai & Rónyai 1990, revised presentation): B = M n (F ), (unknown isomorphism) Every (unital) module is a direct sum of copies of F n V B-module of dimension nr, r = g.c.d(n, r), sr = tn + r, U = B by left mult. find injective φ Hom(U t, V s ), W = im φ V s /W is a B-module of dim. nr. Example: n prime, z any zero divisor in B V = Bz left ideal as module of dimension nr r = 1, construct n-dimensional module isomorphism with M n(f ).
42 Module isomorphism - the general case Reduction to finding minimum size sets of module generators ( surjective morphisms from free modules) H = Hom(V, V ) closed under multiplication: a matrix algebra Hom(V, V ) left H-module if V = V : isomorphism H-mod. generator of Hom(V, V ) the length 1 property of the semisimple case can be exploited to finding min. size sets of generators (I, Karpinski & Saxena (2010)) Surjectivity from free modules Remark: free can be weakened to projective
43 Hidden rank one generators I, Karpinski, Qiao, Santha & Saxena (2014) A = A 0, A 1,..., A k rk A i = 1, but A i unknown (i = 1,..., k) Assume A 0 = ( Ir ), l: smallest s.t. A l ker A 0 im A 0 i 1,..., i l : A il... A i1 ker A 0 im A 0 A l s A ij A s 1 ker A 0 im A 0 when s j For s < j: A ij A s 1 ker A 0 = (0), (otherwise A l j s ker A 0 A il... A ij A s 1 ker A 0 = im A il im A 0 ) For j > s: A l s im A j im A 0, similarly A j : space of solutions for A l s X A s 1 ker A 0 im A 0 (s {1,..., l} \ {j}) system of lin eq.
44 Hidden rank one generators (2) Compute bases for A 1,..., A l A l A 1 ker A 0 A l A 1 ker A 0 im A 0 key property: A il... A i1 im A 0 if i j j for some j Find B i A i : B l B 1 ker A 0 im A 0 Pick a basis element B 1 for A 1 s.t. A l A 2 B 1 ker A 0 im A 0 ; Then B 2 from basis for A 2 s.t. A l A 3 B 2 B 1 ker A 0 im A 0 ; etc.... (B B l ) l ker A 0 = B l... B 1 ker A 0 modulo im A 0 Find λ: λa 0 + B B l has rank rk A 0. rationality issues: rank one generators do not need to be rational example: field extension known rank one generators: over F, even if small
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationAlgebra Exam Syllabus
Algebra Exam Syllabus The Algebra comprehensive exam covers four broad areas of algebra: (1) Groups; (2) Rings; (3) Modules; and (4) Linear Algebra. These topics are all covered in the first semester graduate
More informationALGEBRA QUALIFYING EXAM PROBLEMS
ALGEBRA QUALIFYING EXAM PROBLEMS Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND MODULES General
More informationNote that a unit is unique: 1 = 11 = 1. Examples: Nonnegative integers under addition; all integers under multiplication.
Algebra fact sheet An algebraic structure (such as group, ring, field, etc.) is a set with some operations and distinguished elements (such as 0, 1) satisfying some axioms. This is a fact sheet with definitions
More informationLECTURE 16: REPRESENTATIONS OF QUIVERS
LECTURE 6: REPRESENTATIONS OF QUIVERS IVAN LOSEV Introduction Now we proceed to study representations of quivers. We start by recalling some basic definitions and constructions such as the path algebra
More informationMath 121 Homework 5: Notes on Selected Problems
Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements
More informationSolution. That ϕ W is a linear map W W follows from the definition of subspace. The map ϕ is ϕ(v + W ) = ϕ(v) + W, which is well-defined since
MAS 5312 Section 2779 Introduction to Algebra 2 Solutions to Selected Problems, Chapters 11 13 11.2.9 Given a linear ϕ : V V such that ϕ(w ) W, show ϕ induces linear ϕ W : W W and ϕ : V/W V/W : Solution.
More informationA Little Beyond: Linear Algebra
A Little Beyond: Linear Algebra Akshay Tiwary March 6, 2016 Any suggestions, questions and remarks are welcome! 1 A little extra Linear Algebra 1. Show that any set of non-zero polynomials in [x], no two
More informationProblems in Linear Algebra and Representation Theory
Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific
More informationReal representations
Real representations 1 Definition of a real representation Definition 1.1. Let V R be a finite dimensional real vector space. A real representation of a group G is a homomorphism ρ VR : G Aut V R, where
More informationNONCOMMUTATIVE POLYNOMIAL EQUATIONS. Edward S. Letzter. Introduction
NONCOMMUTATIVE POLYNOMIAL EQUATIONS Edward S Letzter Introduction My aim in these notes is twofold: First, to briefly review some linear algebra Second, to provide you with some new tools and techniques
More informationNotes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers
Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop Eric Sommers 17 July 2009 2 Contents 1 Background 5 1.1 Linear algebra......................................... 5 1.1.1
More informationAlgebra Qualifying Exam August 2001 Do all 5 problems. 1. Let G be afinite group of order 504 = 23 32 7. a. Show that G cannot be isomorphic to a subgroup of the alternating group Alt 7. (5 points) b.
More informationL(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that
ALGEBRAIC GROUPS 61 5. Root systems and semisimple Lie algebras 5.1. Characteristic 0 theory. Assume in this subsection that chark = 0. Let me recall a couple of definitions made earlier: G is called reductive
More informationRings and groups. Ya. Sysak
Rings and groups. Ya. Sysak 1 Noetherian rings Let R be a ring. A (right) R -module M is called noetherian if it satisfies the maximum condition for its submodules. In other words, if M 1... M i M i+1...
More informationGRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.
GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,
More informationTopics in linear algebra
Chapter 6 Topics in linear algebra 6.1 Change of basis I want to remind you of one of the basic ideas in linear algebra: change of basis. Let F be a field, V and W be finite dimensional vector spaces over
More informationMath 121 Homework 4: Notes on Selected Problems
Math 121 Homework 4: Notes on Selected Problems 11.2.9. If W is a subspace of the vector space V stable under the linear transformation (i.e., (W ) W ), show that induces linear transformations W on W
More informationAlgebra Homework, Edition 2 9 September 2010
Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.
More informationSome notes on linear algebra
Some notes on linear algebra Throughout these notes, k denotes a field (often called the scalars in this context). Recall that this means that there are two binary operations on k, denoted + and, that
More information1 Linear Algebra Problems
Linear Algebra Problems. Let A be the conjugate transpose of the complex matrix A; i.e., A = A t : A is said to be Hermitian if A = A; real symmetric if A is real and A t = A; skew-hermitian if A = A and
More informationOnline Exercises for Linear Algebra XM511
This document lists the online exercises for XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Lecture 02 ( 1.1) Online Exercises for Linear Algebra XM511 1) The matrix [3 2
More informationOHSx XM511 Linear Algebra: Solutions to Online True/False Exercises
This document gives the solutions to all of the online exercises for OHSx XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Answers are in square brackets [. Lecture 02 ( 1.1)
More informationThe Graph Isomorphism Problem and the Module Isomorphism Problem. Harm Derksen
The Graph Isomorphism Problem and the Module Isomorphism Problem Harm Derksen Department of Mathematics Michigan Center for Integrative Research in Critical Care University of Michigan Partially supported
More informationMath 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille
Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is
More informationLinear Algebra. Workbook
Linear Algebra Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: November 21 Student: Fall 2011 Checklist Name: A B C D E F F G H I J 1 2 3 4 5 6 7 8 9 10 xxx xxx xxx
More informationLecture Summaries for Linear Algebra M51A
These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture
More informationComputational Approaches to Finding Irreducible Representations
Computational Approaches to Finding Irreducible Representations Joseph Thomas Research Advisor: Klaus Lux May 16, 2008 Introduction Among the various branches of algebra, linear algebra has the distinctions
More informationRing Theory Problems. A σ
Ring Theory Problems 1. Given the commutative diagram α A σ B β A σ B show that α: ker σ ker σ and that β : coker σ coker σ. Here coker σ = B/σ(A). 2. Let K be a field, let V be an infinite dimensional
More informationHonors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime
More informationHomework 5 M 373K Mark Lindberg and Travis Schedler
Homework 5 M 373K Mark Lindberg and Travis Schedler 1. Artin, Chapter 3, Exercise.1. Prove that the numbers of the form a + b, where a and b are rational numbers, form a subfield of C. Let F be the numbers
More informationALGEBRAIC GROUPS J. WARNER
ALGEBRAIC GROUPS J. WARNER Let k be an algebraically closed field. varieties unless otherwise stated. 1. Definitions and Examples For simplicity we will work strictly with affine Definition 1.1. An algebraic
More informationNOTES ON FINITE FIELDS
NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining
More informationNotes on p-divisible Groups
Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete
More informationLINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS
LINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F has characteristic zero. The following are facts (in
More informationCOURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA
COURSE SUMMARY FOR MATH 504, FALL QUARTER 2017-8: MODERN ALGEBRA JAROD ALPER Week 1, Sept 27, 29: Introduction to Groups Lecture 1: Introduction to groups. Defined a group and discussed basic properties
More informationStructure of rings. Chapter Algebras
Chapter 5 Structure of rings 5.1 Algebras It is time to introduce the notion of an algebra over a commutative ring. So let R be a commutative ring. An R-algebra is a ring A (unital as always) together
More informationCHAPTER I. Rings. Definition A ring R is a set with two binary operations, addition + and
CHAPTER I Rings 1.1 Definitions and Examples Definition 1.1.1. A ring R is a set with two binary operations, addition + and multiplication satisfying the following conditions for all a, b, c in R : (i)
More informationA connection between number theory and linear algebra
A connection between number theory and linear algebra Mark Steinberger Contents 1. Some basics 1 2. Rational canonical form 2 3. Prime factorization in F[x] 4 4. Units and order 5 5. Finite fields 7 6.
More informationALGEBRA EXERCISES, PhD EXAMINATION LEVEL
ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)
More informationREPRESENTATION THEORY. WEEKS 10 11
REPRESENTATION THEORY. WEEKS 10 11 1. Representations of quivers I follow here Crawley-Boevey lectures trying to give more details concerning extensions and exact sequences. A quiver is an oriented graph.
More information1.4 Solvable Lie algebras
1.4. SOLVABLE LIE ALGEBRAS 17 1.4 Solvable Lie algebras 1.4.1 Derived series and solvable Lie algebras The derived series of a Lie algebra L is given by: L (0) = L, L (1) = [L, L],, L (2) = [L (1), L (1)
More informationSection 18 Rings and fields
Section 18 Rings and fields Instructor: Yifan Yang Spring 2007 Motivation Many sets in mathematics have two binary operations (and thus two algebraic structures) For example, the sets Z, Q, R, M n (R)
More informationSupplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.
Glossary 1 Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.23 Abelian Group. A group G, (or just G for short) is
More informationRepresentation Theory
Representation Theory Representations Let G be a group and V a vector space over a field k. A representation of G on V is a group homomorphism ρ : G Aut(V ). The degree (or dimension) of ρ is just dim
More information12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold.
12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold. 12.1. Indecomposability of M and the localness of End
More informationRepresentations of quivers
Representations of quivers Gwyn Bellamy October 13, 215 1 Quivers Let k be a field. Recall that a k-algebra is a k-vector space A with a bilinear map A A A making A into a unital, associative ring. Notice
More informationNOTES ON SPLITTING FIELDS
NOTES ON SPLITTING FIELDS CİHAN BAHRAN I will try to define the notion of a splitting field of an algebra over a field using my words, to understand it better. The sources I use are Peter Webb s and T.Y
More information[06.1] Given a 3-by-3 matrix M with integer entries, find A, B integer 3-by-3 matrices with determinant ±1 such that AMB is diagonal.
(January 14, 2009) [06.1] Given a 3-by-3 matrix M with integer entries, find A, B integer 3-by-3 matrices with determinant ±1 such that AMB is diagonal. Let s give an algorithmic, rather than existential,
More informationTHE MINIMAL POLYNOMIAL AND SOME APPLICATIONS
THE MINIMAL POLYNOMIAL AND SOME APPLICATIONS KEITH CONRAD. Introduction The easiest matrices to compute with are the diagonal ones. The sum and product of diagonal matrices can be computed componentwise
More informationLinear and Bilinear Algebra (2WF04) Jan Draisma
Linear and Bilinear Algebra (2WF04) Jan Draisma CHAPTER 3 The minimal polynomial and nilpotent maps 3.1. Minimal polynomial Throughout this chapter, V is a finite-dimensional vector space of dimension
More informationAlgebra Qualifying Exam Solutions. Thomas Goller
Algebra Qualifying Exam Solutions Thomas Goller September 4, 2 Contents Spring 2 2 2 Fall 2 8 3 Spring 2 3 4 Fall 29 7 5 Spring 29 2 6 Fall 28 25 Chapter Spring 2. The claim as stated is false. The identity
More informationNotes on Representations of Finite Groups. Ambar N. Sengupta
Notes on Representations of Finite Groups Ambar N. Sengupta 18th December, 2007 2 Ambar N. Sengupta Contents Preface............................... 6 1 Basic Definitions 7 1.1 Representations of Groups....................
More informationExercises on chapter 1
Exercises on chapter 1 1. Let G be a group and H and K be subgroups. Let HK = {hk h H, k K}. (i) Prove that HK is a subgroup of G if and only if HK = KH. (ii) If either H or K is a normal subgroup of G
More informationRepresentations. 1 Basic definitions
Representations 1 Basic definitions If V is a k-vector space, we denote by Aut V the group of k-linear isomorphisms F : V V and by End V the k-vector space of k-linear maps F : V V. Thus, if V = k n, then
More informationLINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM
LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator
More informationMIT Algebraic techniques and semidefinite optimization May 9, Lecture 21. Lecturer: Pablo A. Parrilo Scribe:???
MIT 6.972 Algebraic techniques and semidefinite optimization May 9, 2006 Lecture 2 Lecturer: Pablo A. Parrilo Scribe:??? In this lecture we study techniques to exploit the symmetry that can be present
More informationIDEAL CLASSES AND RELATIVE INTEGERS
IDEAL CLASSES AND RELATIVE INTEGERS KEITH CONRAD The ring of integers of a number field is free as a Z-module. It is a module not just over Z, but also over any intermediate ring of integers. That is,
More informationRepresentations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III
Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group
More informationQ N id β. 2. Let I and J be ideals in a commutative ring A. Give a simple description of
Additional Problems 1. Let A be a commutative ring and let 0 M α N β P 0 be a short exact sequence of A-modules. Let Q be an A-module. i) Show that the naturally induced sequence is exact, but that 0 Hom(P,
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationMath 210B: Algebra, Homework 4
Math 210B: Algebra, Homework 4 Ian Coley February 5, 2014 Problem 1. Let S be a multiplicative subset in a commutative ring R. Show that the localisation functor R-Mod S 1 R-Mod, M S 1 M, is exact. First,
More informationALGEBRA QUALIFYING EXAM SPRING 2012
ALGEBRA QUALIFYING EXAM SPRING 2012 Work all of the problems. Justify the statements in your solutions by reference to specific results, as appropriate. Partial credit is awarded for partial solutions.
More information22M: 121 Final Exam. Answer any three in this section. Each question is worth 10 points.
22M: 121 Final Exam This is 2 hour exam. Begin each question on a new sheet of paper. All notations are standard and the ones used in class. Please write clearly and provide all details of your work. Good
More informationALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS.
ALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS. KEVIN MCGERTY. 1. RINGS The central characters of this course are algebraic objects known as rings. A ring is any mathematical structure where you can add
More informationSince G is a compact Lie group, we can apply Schur orthogonality to see that G χ π (g) 2 dg =
Problem 1 Show that if π is an irreducible representation of a compact lie group G then π is also irreducible. Give an example of a G and π such that π = π, and another for which π π. Is this true for
More informationMODEL ANSWERS TO THE FIRST HOMEWORK
MODEL ANSWERS TO THE FIRST HOMEWORK 1. Chapter 4, 1: 2. Suppose that F is a field and that a and b are in F. Suppose that a b = 0, and that b 0. Let c be the inverse of b. Multiplying the equation above
More informationLECTURE NOTES AMRITANSHU PRASAD
LECTURE NOTES AMRITANSHU PRASAD Let K be a field. 1. Basic definitions Definition 1.1. A K-algebra is a K-vector space together with an associative product A A A which is K-linear, with respect to which
More informationGEOMETRIC CONSTRUCTIONS AND ALGEBRAIC FIELD EXTENSIONS
GEOMETRIC CONSTRUCTIONS AND ALGEBRAIC FIELD EXTENSIONS JENNY WANG Abstract. In this paper, we study field extensions obtained by polynomial rings and maximal ideals in order to determine whether solutions
More informationAlgebra I Fall 2007
MIT OpenCourseWare http://ocw.mit.edu 18.701 Algebra I Fall 007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.701 007 Geometry of the Special Unitary
More information1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det
What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix
More informationRings. EE 387, Notes 7, Handout #10
Rings EE 387, Notes 7, Handout #10 Definition: A ring is a set R with binary operations, + and, that satisfy the following axioms: 1. (R, +) is a commutative group (five axioms) 2. Associative law for
More informationAlgebra Review. Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor. June 15, 2001
Algebra Review Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor June 15, 2001 1 Groups Definition 1.1 A semigroup (G, ) is a set G with a binary operation such that: Axiom 1 ( a,
More informationOn the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem
On the singular elements of a semisimple Lie algebra and the generalized Amitsur-Levitski Theorem Bertram Kostant, MIT Conference on Representations of Reductive Groups Salt Lake City, Utah July 10, 2013
More informationHodge Structures. October 8, A few examples of symmetric spaces
Hodge Structures October 8, 2013 1 A few examples of symmetric spaces The upper half-plane H is the quotient of SL 2 (R) by its maximal compact subgroup SO(2). More generally, Siegel upper-half space H
More informationORAL QUALIFYING EXAM QUESTIONS. 1. Algebra
ORAL QUALIFYING EXAM QUESTIONS JOHN VOIGHT Below are some questions that I have asked on oral qualifying exams (starting in fall 2015). 1.1. Core questions. 1. Algebra (1) Let R be a noetherian (commutative)
More informationGraduate Preliminary Examination
Graduate Preliminary Examination Algebra II 18.2.2005: 3 hours Problem 1. Prove or give a counter-example to the following statement: If M/L and L/K are algebraic extensions of fields, then M/K is algebraic.
More informationSUMMARY ALGEBRA I LOUIS-PHILIPPE THIBAULT
SUMMARY ALGEBRA I LOUIS-PHILIPPE THIBAULT Contents 1. Group Theory 1 1.1. Basic Notions 1 1.2. Isomorphism Theorems 2 1.3. Jordan- Holder Theorem 2 1.4. Symmetric Group 3 1.5. Group action on Sets 3 1.6.
More informationPiecewise Noetherian Rings
Northern Illinois University UNAM 25 May, 2017 Acknowledgments Results for commutative rings are from two joint papers with William D. Weakley,, Comm. Algebra (1984) and A note on prime ideals which test
More informationMATH 8253 ALGEBRAIC GEOMETRY WEEK 12
MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f
More informationMath 120 HW 9 Solutions
Math 120 HW 9 Solutions June 8, 2018 Question 1 Write down a ring homomorphism (no proof required) f from R = Z[ 11] = {a + b 11 a, b Z} to S = Z/35Z. The main difficulty is to find an element x Z/35Z
More informationSome properties of index of Lie algebras
arxiv:math/0001042v1 [math.rt] 7 Jan 2000 Some properties of index of Lie algebras Contents Vladimir Dergachev February 1, 2008 1 Introduction 1 2 Index of Lie algebras 2 3 Tensor products 3 4 Associative
More informationLinear Algebra. Min Yan
Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................
More informationThus, the integral closure A i of A in F i is a finitely generated (and torsion-free) A-module. It is not a priori clear if the A i s are locally
Math 248A. Discriminants and étale algebras Let A be a noetherian domain with fraction field F. Let B be an A-algebra that is finitely generated and torsion-free as an A-module with B also locally free
More informationElementary linear algebra
Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The
More informationOhio State University Department of Mathematics Algebra Qualifier Exam Solutions. Timothy All Michael Belfanti
Ohio State University Department of Mathematics Algebra Qualifier Exam Solutions Timothy All Michael Belfanti July 22, 2013 Contents Spring 2012 1 1. Let G be a finite group and H a non-normal subgroup
More informationINTRODUCTION TO LIE ALGEBRAS. LECTURE 2.
INTRODUCTION TO LIE ALGEBRAS. LECTURE 2. 2. More examples. Ideals. Direct products. 2.1. More examples. 2.1.1. Let k = R, L = R 3. Define [x, y] = x y the cross-product. Recall that the latter is defined
More informationREPRESENTATION THEORY, LECTURE 0. BASICS
REPRESENTATION THEORY, LECTURE 0. BASICS IVAN LOSEV Introduction The aim of this lecture is to recall some standard basic things about the representation theory of finite dimensional algebras and finite
More informationMultivariate Public Key Cryptography or Why is there a rainbow hidden behind fields full of oil and vinegar?
Multivariate Public Key Cryptography or Why is there a rainbow hidden behind fields full of oil and vinegar? Christian Eder, Jean-Charles Faugère and Ludovic Perret Seminar on Fundamental Algorithms, University
More informationMath 210B. Artin Rees and completions
Math 210B. Artin Rees and completions 1. Definitions and an example Let A be a ring, I an ideal, and M an A-module. In class we defined the I-adic completion of M to be M = lim M/I n M. We will soon show
More informationMODEL ANSWERS TO HWK #7. 1. Suppose that F is a field and that a and b are in F. Suppose that. Thus a = 0. It follows that F is an integral domain.
MODEL ANSWERS TO HWK #7 1. Suppose that F is a field and that a and b are in F. Suppose that a b = 0, and that b 0. Let c be the inverse of b. Multiplying the equation above by c on the left, we get 0
More informationMAT 445/ INTRODUCTION TO REPRESENTATION THEORY
MAT 445/1196 - INTRODUCTION TO REPRESENTATION THEORY CHAPTER 1 Representation Theory of Groups - Algebraic Foundations 1.1 Basic definitions, Schur s Lemma 1.2 Tensor products 1.3 Unitary representations
More informationList of topics for the preliminary exam in algebra
List of topics for the preliminary exam in algebra 1 Basic concepts 1. Binary relations. Reflexive, symmetric/antisymmetryc, and transitive relations. Order and equivalence relations. Equivalence classes.
More informationABSTRACT NONSINGULAR CURVES
ABSTRACT NONSINGULAR CURVES Affine Varieties Notation. Let k be a field, such as the rational numbers Q or the complex numbers C. We call affine n-space the collection A n k of points P = a 1, a,..., a
More information1 Fields and vector spaces
1 Fields and vector spaces In this section we revise some algebraic preliminaries and establish notation. 1.1 Division rings and fields A division ring, or skew field, is a structure F with two binary
More informationALGEBRA PH.D. QUALIFYING EXAM September 27, 2008
ALGEBRA PH.D. QUALIFYING EXAM September 27, 2008 A passing paper consists of four problems solved completely plus significant progress on two other problems; moreover, the set of problems solved completely
More informationGeneralized eigenspaces
Generalized eigenspaces November 30, 2012 Contents 1 Introduction 1 2 Polynomials 2 3 Calculating the characteristic polynomial 5 4 Projections 7 5 Generalized eigenvalues 10 6 Eigenpolynomials 15 1 Introduction
More informationA classification of sharp tridiagonal pairs. Tatsuro Ito, Kazumasa Nomura, Paul Terwilliger
Tatsuro Ito Kazumasa Nomura Paul Terwilliger Overview This talk concerns a linear algebraic object called a tridiagonal pair. We will describe its features such as the eigenvalues, dual eigenvalues, shape,
More informationDefinition 9.1. The scheme µ 1 (O)/G is called the Hamiltonian reduction of M with respect to G along O. We will denote by R(M, G, O).
9. Calogero-Moser spaces 9.1. Hamiltonian reduction along an orbit. Let M be an affine algebraic variety and G a reductive algebraic group. Suppose M is Poisson and the action of G preserves the Poisson
More informationChapter 2 Linear Transformations
Chapter 2 Linear Transformations Linear Transformations Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more
More information