Semigroup of generalized inverses of matrices


 Lewis Booker
 4 years ago
 Views:
Transcription
1 Semigroup of generlized inverses of mtrices Hnif Zekroui nd Sid Guedjib Abstrct. The pper is divided into two principl prts. In the first one, we give the set of generlized inverses of mtrix A structure of semigroup nd study some lgebric properties like fctoriztion nd commuttivity. We lso define n equivlence reltion in order to estblish n isomorphism between the quotient semigroup nd the semigroup of projectors on RA. In the second prt, we study reltion between semigroups ssocited to equivlent mtrices nd estblish correspondence between the set of mtrices nd the set of ssocited semigroups. We lso study some lgebric properties in this set like intersection nd prtil order. M.S.C. 2000: 15A09, 15A24, 20M15. Key words: Generlized inverse; equivlent mtrices; projectors; prtil order. 1 Introduction In the following, K represents the rel or the complex field. Let A be n m n mtrix over K. The generlized inverse, or the {1} inverse of A is n n m mtrix X over K, which stisfies the mtrix eqution AXA = A. If in ddition X stisfies the eqution XAX = X, then X is sid to be reflexive generlized inverse or {1, 2} inverse of A. It is known tht there re infinitely sets of {1} inverses nd {1, 2} inverses of mtrix A. Mny studies on generlized inverses nd their pplictions hve been done see [2], [5], [1]. Some of them re lgebric; like the sum see [4] nd the reverse order lw of product see [7]. In some res of binry mtrices, the usge of the generlized inverse of mtrix might provide results s well, like the CP property for binry mtrices see [5]. In this pper we will study two principl prts. In the first one, we will give the set of {1} inverses of mtrix A structure of semigroup nd study some lgebric properties like fctoriztion nd commuttivity. To study the property of fctoriztion in these semigroups, we will use the MoorePenrose inverse. We will lso define n equivlence reltion in order to estblish n isomorphism between the quotient semigroup nd the semigroup of projectors on RA. The second prt concerns the reltion between semigroups ssocited to equivlent mtrices nd the Applied Sciences, Vol.12, 2010, pp c Blkn Society of Geometers, Geometry Blkn Press 2010.
2 Semigroup of generlized inverses of mtrices 147 correspondence between the set M m n K nd the set of the ssocited semigroups. Some lgebric properties in this set will be treted; like the intersection of semigroups nd isomorphisms. Also, ordering of sets in the rel inner product spce, is recent subject. In [7], this subject ddressed s min result, nd proved using the LorentzMinkowski distnce for ordering certin sets. We shll lso follow this pttern by using generlized inverses for ordering sets of the introduced semigroups by minus prtil order. 2 Preliminry notes Definition 2.1. Let A be n m n mtrix over K. The MoorePenrose inverse of A denoted by A + is the n m mtrix X over K which stisfies the four equtions 2.1 AXA = A, XAX = X, AX = AX, XA = XA. For every mtrix there exists only one MoorePenrose inverse. If X is t lest {1} inverse of A, then AX nd XA re projectors on RA nd RX the rnge spces of A nd X respectively nd rnkax = rnka = rnkxa. For more properties nd remrks, see [2], [1]. Denote by A {1} nd by A {1,2} the sets of ll {1} inverses nd {1, 2} inverses of A respectively. We will denote by smll letters the submtrices of mtrix X nd by I nd 0 the identicl nd zero mtrices or identicl nd zero submtrices. Lemms without proofs re either esy to proof or well known in the precedent references. Lemm 2.1. Let A be n m n mtrix over K of rnk r. Then 1 There exist non singulr mtrices P nd Q such tht A = Q 1 2 A + = P 1 1 Q. 3 The elements of A {1} re of the form P 1 1 r f A {1,2} re of the form P 1 1 r f e f r e Q. e g P. Q nd the elements of The first ssertion is n elementry lemm in liner lgebr. The second one nd the third need only direct verifiction in 2.1. For this purpose the proof ws omitted.
3 148 Hnif Zekroui nd Sid Guedjib 3 Min results 3.1 Semigroup on A {1} Fctoriztion nd commuttivity The min importnt point in the theorem below is fctoriztion of A {1,2}. For this purpose we introduce two clsses of A {1} : { } P A+ A = X A {1} /X = P 1 1 Q P AA + = { X A {1} /X = P 1 1 x 0 } Q. The nottions P A + A nd P AA + re justified by the fct tht these two clsses re the sets of fixed points under left nd right multiplictions by orthogonl projectors A + A nd AA +. In fct, for ny X A {1}, A + AX = X X = P 1 1 Q nd XAA + = X X = P 1 1 Q. x 0 Theorem 3.1. Let be lw on A {1} defined s follows: for ny X, Y A {1} ; X Y = XAY. Then 1 A {1} is semigroup. 2 For ny X, Y A {1} we hve X Y A {1,2} 3 A {1,2} is n idel of A {1}. 4 P AA + P A + A = A {1,2} nd P AA + P A + A = {A + }. 5 For ny X nd Y in A {1} we hve X Y = Y X AX = AY nd XA = Y A. Proof. 1 A XAY A = AXA Y A = AY A = A. Then X Y = XAY A {1}. The ssocitivity of is obtined from the ssocitivity of product of mtrices. So A {1} is semigroup. 2 Let X nd Y be in A {1}. Since A XAY A = A nd XAY A XAY = X AY A XAY = X AXA Y = XAY, we obtin X Y = XAY A {1,2}. 3 As A {1,2} A {1}, then 3 is just simple consequence of 2. 4 First, remrk tht P AA + nd P A + A re subsemigroups of A {1,2}. Let A = Q 1 P. For ny Z = P 1 1 Q A y y {1,2} there exist Y = P 1 1 Q P y 0 AA + nd X = P 1 1 Q P A + A such tht Y X = P Q y 0 1 = P 1 1 Q = Z. Consequently, we hve P AA + P A + A = A {1,2}. However, by direct computtion, we y y fined tht P A + A P AA + = {A + 1 }. For Z = p Q y y 1 P AA + P A + A we hve x = 0 nd y = 0. Then Z = P 1 Q 1 = A +. Therefore P AA + P A + A = {A + }. This ssertion cn be replced by uniqueness of fctoriztion of elements of A {1,2}. 5 For ny X nd Y in A {1} we hve XAY = Y AX AXAY =
4 Semigroup of generlized inverses of mtrices 149 AY AX AY = AX nd XAY = Y AX XAY A = Y AXA XA = Y A. Conversely, AY = AX XAY = XAX nd Y A = XA Y AX = XAX. Hence we hve X Y = Y X AX = AY nd XA = Y A. Note tht we hve not relly commuttivity on A {1}, but we hve kind of commuttivity modulo projectors. It is esy to prove tht the set of projectors of vector spce on subspce is semigroup under ordinry composition. As mny studies on projectors nd their properties hve been done, we will exploit this to mke properties of A {1} more esy. This clim will be reched by defining n equivlence reltion in A {1}, thus we will hve n isomorphism between the quotient semigroup of A {1} nd the semigroup of projectors on RA. It will cuse no confusion if we use the sme letter to designte projector nd its ssocited mtrix. Theorem 3.2. Let A M m n K, Π be the semigroup of projectors of K m on R A. Then we hve 1 For ny P Π, there exists X A {1} such tht AX = P. 2 There is n equivlence reltion on A {1} such tht under the quotient lw of, is semigroup isomorphic to Π. A {1} Proof. 1 Let P Π nd A A {1}. As AA is projector on R A, we hve AA P = P nd P A = A. Thus if we tke X = A P, we hve AXA = AA P A = AA A = A nd AX = P which mens tht Π = { AX/X A {1}}. 2 Let be reltion in A {1} defined by X Y AX = AY. Then it is esy to check tht is n equivlence reltion in A {1}. Let χ be the cnonicl mp from A {1} on A{1} A{1}. Then for every χx, χy, the quotient lw of is defined by χxχ Y = χ X Y. It esy to show tht A{1} is semigroup nd χ is n homomorphism. It immeditely follows tht there exists mp ψ from A {1} to Π defined by ψχx = AX. Now we check tht ψ is n homomorphism. Since A XAY = AXA Y = AY, we obtin XAY Y. Then χ XAY = χy. Therefore ψ χxχy = ψ χ X Y = ψ χ XAY = ψ χy = AY = AXA Y = = AX AY = ψ χ X ψ χy. Now, since AX = AY, it follows tht χx = χy. We conclude tht for ny AX Π there is unique χx A{1} such tht ψχx = AX which mens tht ψ is bijection. 3.2 The set A {1} of semigroups Denote by M {1} m nk the set M {1} m nk = { A {1} /A M m n K }. In this subsection, we will estblish reltion between M m n K nd M m nk {1} nd deduce some properties of M m nk {1} like isomorphism of semigroups, intersection of them nd reltion order. For this min, the following Lemm will be useful. Lemm 3.1. [2], [1]. Let A nd B be two equivlent mtrices, such tht B = Q 1 AP. Then, for every Y in B {1} there exists unique X in A {1} such tht Y = P 1 XQ.
5 150 Hnif Zekroui nd Sid Guedjib Isomorphism between semigroups Theorem 3.3. Let A nd B be two equivlent mtrices. Then A {1}, nd B {1}, re isomorphic. Proof. By using the previous Lemm, we cn define mp ϕ from A {1} on B {1} s follows ϕx = P 1 XQ. Then ϕ 1 is the inverse mp from B {1} on A {1} given by ϕ 1 X = P XQ 1. In ddition, for every X nd Y in A {1}, we hve ϕx Y = ϕxay = P 1 XAY Q = P 1 XQQ 1 AP P 1 Y Q = = ϕxbϕy = ϕx ϕy. Also, we hve for every X nd Y in B {1}, ϕ 1 X Y = ϕ 1 XBY = ϕ 1 XAϕ 1 Y = ϕ 1 X ϕ 1 Y. Then the mp ϕ is n isomorphism. We remrk tht ϕ A + = P 1 A + Q = B + only if P nd Q re orthogonl. Lemm 3.2. [4] Let A nd B be two mtrices. Then the following sttements re equivlent: rnka + rnkb A = rnkb. b Every {1} inverse of B is {1} inverse of both A nd B A. c R A R B = {0} nd R A t R B t = {0}. Theorem 3.4. There is onetoone correspondence between M m n K nd M {1} m nk mps 0 to M n m K nd preserves isomorphisms between semigroups. Proof. Let ψ be mp from M m n K onto M m nk {1} defined for every A M m n K by ψa = A {1}. Since 0X0 = 0 for ny X M n m K, we get 0 {1} = M n m K. Thus ψ0 = M n m K. According to Lemm 3.2, if A {1} = B {1}, we hve rnka + rnkb A = rnkb nd rnkb + rnka B = rnka. Thus we hve rnk A B = 0 = rnk B A. Therefore A = B. Now, let A nd B M m n K such tht B = Q 1 AP. According to Lemm 3.1 nd Theorem 3.3, we hve B {1} = { P 1 XQ/X A {1}} = ϕ A {1}. Hence we hve ψb = ϕψa Intersection of semigroups Theorem 3.5. For every mtrix A M m n K there exists mtrix A M m n K such tht A {1} A {1}. b For ny mtrices A, B M m n K there exists n isomorphism ϕ from A {1} on ϕ A {1} such tht ϕ A {1} B {1}. Proof. Let rnka = r min m, n. It is sufficient to prove tht for A M m n K with rnka = r, there exists mtrix A M m n K such tht rnk A + A = rnka + rnka = min m, n. Applying Lemm 3.2, we hve every {1} inverse of A + A is {1} inverse of both A nd A. Thus A {1} A {1}. Let P nd
6 Semigroup of generlized inverses of mtrices 151 Q be two nonsingulr mtrices such tht A = Q 1 I P. Then it is sufficient to tke A = Q 1 P with w M 0 w m r n r K nd rnkw = min m, n r. b Let rnka = r, rnkb = s. According to 1, there exist mtrices A nd B M m n K of rnks min m, n r nd min m, n s such tht rnk A + A = rnka + rnka = min m, n nd rnk B + B = rnkb + rnkb = min m, n. It follows tht A + A {1} A {1} A {1}, B + B {1} B {1} B {1}. Since A + A, B + B hve the sme rnk, they re equivlent. So, there exists n isomorphism ϕ from A {1} on ϕ A {1} such tht ϕ A + A {1} = B + B {1}. Hence we hve ϕ A + A {1} ϕ A {1} A {1} = ϕ A {1} ϕ A {1} nd ϕ A + A {1} B {1} B {1}. Finlly, we hve ϕ A {1} B {1}. The nturl question rises whether it is possible to conclude tht for ny mtrices A, B M m n K we hve A {1} B {1}. The nswer is negtive. In fct if we tke B = αa for sclr α which is different of 1 nd 0, then if X A {1} B {1}, we hve αa = αaxαa = α 2 A. Thus α {0, 1} which contrdicts our ssumption. Consequently we hve A {1} B {1} = Prtil order in M {1} m n K Definition 3.1. [1] A minus prtil order, denoted by, is defined s follows for A, B M m n K, then A B if rnkb = rnka + rnk B A. Theorem The inclusion is prtil order in M m nk {} induced by the minus order in the reverse order. b Let = min m, n. For ny mtrix A M m n K of rnk r there exists sequence of mtrices A = A r A r+1... A m0 in M m n K such tht rnk A r = rnk A = r, rnka r+i = r + i for i = 1,... r. Thus, there exists sequence A {1} A {1} r = A {1} nd A {1} is the lst term. Proof. For A, B M m n K, if A B, then rnkb = rnka+rnk B A. By Lemm 3.2 it follows tht B {1} A {1}. Hence we hve the prtil order in M m nk. {1} b Let {v 1, v 2,...v r } be bsis of RA nd {v r+1, v r+2,...v m } be such together with the bsis of RA form bsis for K m. Let {e 1, e 2,...e n } be bsis for K n such tht A r e j = Ae j = v j for j = 1,...r nd A r e j = 0 for j = r + 1,...m nd for i = 1,... r, A r+i e j = v j for j = 1,...r + i nd A r+i e j = 0 for j = r + i + 1,...m. In these bses the mtrices A = A r nd A r+i re of the form A = A r = Q 1 I P, A r+i = Q 1 Ir+i 0 P for i = 1,...m 0 r. Then we hve rnk A r = rnka = r, nd rnk A r+i = rnk A r + i = r + i = rnka r + rnk A r+i A r,
7 152 Hnif Zekroui nd Sid Guedjib for i = 1,... r. Thus A = A r A r+1... A m0. By, we hve A {1}... A {1} r = A {1}. A {1} is the lst term becuse A m0 is of mximl rnk. 4 Conclusions In the present pper we give the set of generlized inverses of mtrix structure of semigroup nd fctorize its elements to simple ones. Unfortuntely, this semigroup is not commuttive. To get nice structurl result, we define n equivlence reltion nd we obtin n isomorphism between the quotient semigroup nd the semigroup of projectors. Furthermore we estblish onetoone correspondence between the set of mtrices nd the set of ssocited semigroups. This correspondence preserves isomorphisms between semigroups nd mps the zero mtrix which is the zero of the dditive group of mtrices into the entire set of mtrices which is the unity relted to intersection of sets. Finlly, we hve proved tht for ny mtrix, there exists sequence of semigroups ordered by inclusion. References [1] R.B. Bpt, Liner Algebr nd Liner Models, Springer [2] BenIsrel nd T.T. Greville, Generlized inverses, theory nd pplictions, Kreiger Eds [3] O. Demirel, E. Soyturk, On the ordered sets in ndimensionl rel inner product spces, Applied Sciences , [4] J.A. Fill nd D.E. Fishkind, The MoorePenrose Generlized Inverse for Sums of Mtrices, SIAM J. Mtrix Anl. Appl , [5] M.Z. Nshed, Generlized Inverses, Theory nd Applictions, Acdemic Press [6] L. Noui, A note on consecutive ones in binry mtrix, Applied Sciences , [7] Z. Xiong, B. Zheng, The reverse order lws for {1, 2, 3} nd {1, 2, 4} inverses of twomtrix product, Appl. Mth. Lett. 21, , Authors ddress: Hnif Zekroui nd Sid Guedjib Deprtment of Mthemtics, Fculty of Sciences, University of Btn, 05000, Btn, Algeri. Emil:
Theoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationHere we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.
Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous
More informationad = cb (1) cf = ed (2) adf = cbf (3) cf b = edb (4)
10 Most proofs re left s reding exercises. Definition 10.1. Z = Z {0}. Definition 10.2. Let be the binry reltion defined on Z Z by, b c, d iff d = cb. Theorem 10.3. is n equivlence reltion on Z Z. Proof.
More informationElements of Matrix Algebra
Elements of Mtrix Algebr Klus Neusser Kurt Schmidheiny September 30, 2015 Contents 1 Definitions 2 2 Mtrix opertions 3 3 Rnk of Mtrix 5 4 Specil Functions of Qudrtic Mtrices 6 4.1 Trce of Mtrix.........................
More informationResults on Planar Near Rings
Interntionl Mthemticl Forum, Vol. 9, 2014, no. 23, 11391147 HIKARI Ltd, www.mhikri.com http://dx.doi.org/10.12988/imf.2014.4593 Results on Plnr Ner Rings Edurd Domi Deprtment of Mthemtics, University
More informationLecture 2e Orthogonal Complement (pages )
Lecture 2e Orthogonl Complement (pges ) We hve now seen tht n orthonorml sis is nice wy to descrie suspce, ut knowing tht we wnt n orthonorml sis doesn t mke one fll into our lp. In theory, the process
More informationN 0 completions on partial matrices
N 0 completions on prtil mtrices C. Jordán C. Mendes Arújo Jun R. Torregros Instituto de Mtemátic Multidisciplinr / Centro de Mtemátic Universidd Politécnic de Vlenci / Universidde do Minho Cmino de Ver
More informationMatrices. Elementary Matrix Theory. Definition of a Matrix. Matrix Elements:
Mtrices Elementry Mtrix Theory It is often desirble to use mtrix nottion to simplify complex mthemticl expressions. The simplifying mtrix nottion usully mkes the equtions much esier to hndle nd mnipulte.
More informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
More informationarxiv: v1 [math.ra] 1 Nov 2014
CLASSIFICATION OF COMPLEX CYCLIC LEIBNIZ ALGEBRAS DANIEL SCOFIELD AND S MCKAY SULLIVAN rxiv:14110170v1 [mthra] 1 Nov 2014 Abstrct Since Leibniz lgebrs were introduced by Lody s generliztion of Lie lgebrs,
More informationMATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35
MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35 9. Modules over PID This week we re proving the fundmentl theorem for finitely generted modules over PID, nmely tht they re ll direct sums of cyclic modules.
More informationTHE QUADRATIC RECIPROCITY LAW OF DUKEHOPKINS. Circa 1870, G. Zolotarev observed that the Legendre symbol ( a p
THE QUADRATIC RECIPROCITY LAW OF DUKEHOPKINS PETE L CLARK Circ 1870, Zolotrev observed tht the Legendre symbol ( p ) cn be interpreted s the sign of multipliction by viewed s permuttion of the set Z/pZ
More informationON THE NILPOTENCY INDEX OF THE RADICAL OF A GROUP ALGEBRA. XI
Mth. J. Okym Univ. 44(2002), 51 56 ON THE NILPOTENCY INDEX OF THE RADICAL OF A GROUP ALGEBRA. XI Koru MOTOSE Let t(g) be the nilpotency index of the rdicl J(KG) of group lgebr KG of finite psolvble group
More informationHilbert Spaces. Chapter Inner product spaces
Chpter 4 Hilbert Spces 4.1 Inner product spces In the following we will discuss both complex nd rel vector spces. With L denoting either R or C we recll tht vector spce over L is set E equipped with ddition,
More informationECON 331 Lecture Notes: Ch 4 and Ch 5
Mtrix Algebr ECON 33 Lecture Notes: Ch 4 nd Ch 5. Gives us shorthnd wy of writing lrge system of equtions.. Allows us to test for the existnce of solutions to simultneous systems. 3. Allows us to solve
More informationElementary Linear Algebra
Elementry Liner Algebr Anton & Rorres, 1 th Edition Lecture Set 5 Chpter 4: Prt II Generl Vector Spces 163 คณ ตศาสตร ว ศวกรรม 3 สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา 1/2555 163 คณตศาสตรวศวกรรม 3 สาขาวชาวศวกรรมคอมพวเตอร
More informationNumerical Linear Algebra Assignment 008
Numericl Liner Algebr Assignment 008 Nguyen Qun B Hong Students t Fculty of Mth nd Computer Science, Ho Chi Minh University of Science, Vietnm emil. nguyenqunbhong@gmil.com blog. http://hongnguyenqunb.wordpress.com
More informationMTH 5102 Linear Algebra Practice Exam 1  Solutions Feb. 9, 2016
Nme (Lst nme, First nme): MTH 502 Liner Algebr Prctice Exm  Solutions Feb 9, 206 Exm Instructions: You hve hour & 0 minutes to complete the exm There re totl of 6 problems You must show your work Prtil
More informationMath 4310 Solutions to homework 1 Due 9/1/16
Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1
More informationMathCity.org Merging man and maths
MthCity.org Merging mn nd mths Exercise.8 (s) Pge 46 Textbook of Algebr nd Trigonometry for Clss XI Avilble online @ http://, Version: 3.0 Question # Opertion performed on the twomember set G = {0, is
More informationMatrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24
Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the
More informationSTUDY GUIDE FOR BASIC EXAM
STUDY GUIDE FOR BASIC EXAM BRYON ARAGAM This is prtil list of theorems tht frequently show up on the bsic exm. In mny cses, you my be sked to directly prove one of these theorems or these vrints. There
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More informationSTRUCTURED TRIANGULAR LIMIT ALGEBRAS
STRUCTURED TRIANGULAR LIMIT ALGEBRAS DAVID R. LARSON nd BARUCH SOLEL [Received 5 Februry 1996 Revised 24 September 1996] 1. Introduction Suppose is unitl Bnch lgebr which contins n incresing chin n n 1
More informationREPRESENTATION THEORY OF PSL 2 (q)
REPRESENTATION THEORY OF PSL (q) YAQIAO LI Following re notes from book [1]. The im is to show the qusirndomness of PSL (q), i.e., the group hs no low dimensionl representtion. 1. Representtion Theory
More informationIntroduction to Group Theory
Introduction to Group Theory Let G be n rbitrry set of elements, typiclly denoted s, b, c,, tht is, let G = {, b, c, }. A binry opertion in G is rule tht ssocites with ech ordered pir (,b) of elements
More informationDually quaside Morgan Stone semiheyting algebras II. Regularity
Volume 2, Number, July 204, 6582 ISSN Print: 23455853 Online: 2345586 Dully quside Morgn Stone semiheyting lgebrs II. Regulrity Hnmntgoud P. Snkppnvr Abstrct. This pper is the second of two prt series.
More informationNatural examples of rings are the ring of integers, a ring of polynomials in one variable, the ring
More generlly, we define ring to be nonempty set R hving two binry opertions (we ll think of these s ddition nd multipliction) which is n Abelin group under + (we ll denote the dditive identity by 0),
More informationHQPD  ALGEBRA I TEST Record your answers on the answer sheet.
HQPD  ALGEBRA I TEST Record your nswers on the nswer sheet. Choose the best nswer for ech. 1. If 7(2d ) = 5, then 14d 21 = 5 is justified by which property? A. ssocitive property B. commuttive property
More informationContents. Outline. Structured Rank Matrices Lecture 2: The theorem Proofs Examples related to structured ranks References. Structure Transport
Contents Structured Rnk Mtrices Lecture 2: Mrc Vn Brel nd Rf Vndebril Dept. of Computer Science, K.U.Leuven, Belgium Chemnitz, Germny, 2630 September 2011 1 Exmples relted to structured rnks 2 2 / 26
More informationKRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION
Fixed Point Theory, 13(2012), No. 1, 285291 http://www.mth.ubbcluj.ro/ nodecj/sfptcj.html KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION FULI WANG AND FENG WANG School of Mthemtics nd
More informationON THE CINTEGRAL BENEDETTO BONGIORNO
ON THE CINTEGRAL BENEDETTO BONGIORNO Let F : [, b] R be differentible function nd let f be its derivtive. The problem of recovering F from f is clled problem of primitives. In 1912, the problem of primitives
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
More informationFrobenius numbers of generalized Fibonacci semigroups
Frobenius numbers of generlized Fiboncci semigroups Gretchen L. Mtthews 1 Deprtment of Mthemticl Sciences, Clemson University, Clemson, SC 296340975, USA gmtthe@clemson.edu Received:, Accepted:, Published:
More informationSeptember 13 Homework Solutions
College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are
More informationResearch Article Moment Inequalities and Complete Moment Convergence
Hindwi Publishing Corportion Journl of Inequlities nd Applictions Volume 2009, Article ID 271265, 14 pges doi:10.1155/2009/271265 Reserch Article Moment Inequlities nd Complete Moment Convergence Soo Hk
More informationLYAPUNOVTYPE INEQUALITIES FOR THIRDORDER LINEAR DIFFERENTIAL EQUATIONS
Electronic Journl of Differentil Equtions, Vol. 2017 (2017), No. 139, pp. 1 14. ISSN: 10726691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu LYAPUNOVTYPE INEQUALITIES FOR THIRDORDER LINEAR
More informationIntuitionistic Fuzzy Lattices and Intuitionistic Fuzzy Boolean Algebras
Intuitionistic Fuzzy Lttices nd Intuitionistic Fuzzy oolen Algebrs.K. Tripthy #1, M.K. Stpthy *2 nd P.K.Choudhury ##3 # School of Computing Science nd Engineering VIT University Vellore632014, TN, Indi
More information1 The Lagrange interpolation formula
Notes on Qudrture 1 The Lgrnge interpoltion formul We briefly recll the Lgrnge interpoltion formul. The strting point is collection of N + 1 rel points (x 0, y 0 ), (x 1, y 1 ),..., (x N, y N ), with x
More informationBases for Vector Spaces
Bses for Vector Spces 22625 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More informationLinear Algebra 1A  solutions of ex.4
Liner Algebr A  solutions of ex.4 For ech of the following, nd the inverse mtrix (mtritz hofkhit if it exists  ( 6 6 A, B (, C 3, D, 4 4 ( E i, F (inverse over C for F. i Also, pick n invertible mtrix
More informationJOURNÉES ÉQUATIONS AUX DÉRIVÉES PARTIELLES
JOURNÉES ÉQUATIONS AUX DÉRIVÉES PARTIELLES ANDERS MELIN On the construction of fundmentl solutions for differentil opertors on nilpotent groups Journées Équtions ux dérivées prtielles (1981), p. 15
More informationPart IB Numerical Analysis
Prt IB Numericl Anlysis Theorems with proof Bsed on lectures by G. Moore Notes tken by Dexter Chu Lent 2016 These notes re not endorsed by the lecturers, nd I hve modified them (often significntly) fter
More informationREGULARITY OF NONLOCAL MINIMAL CONES IN DIMENSION 2
EGULAITY OF NONLOCAL MINIMAL CONES IN DIMENSION 2 OVIDIU SAVIN AND ENICO VALDINOCI Abstrct. We show tht the only nonlocl sminiml cones in 2 re the trivil ones for ll s 0, 1). As consequence we obtin tht
More informationBest Approximation in the 2norm
Jim Lmbers MAT 77 Fll Semester 111 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
More informationThe Algebra (aljabr) of Matrices
Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (ljbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense
More informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 25pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
More informationIntroduction to Determinants. Remarks. Remarks. The determinant applies in the case of square matrices
Introduction to Determinnts Remrks The determinnt pplies in the cse of squre mtrices squre mtrix is nonsingulr if nd only if its determinnt not zero, hence the term determinnt Nonsingulr mtrices re sometimes
More informationVariational Techniques for SturmLiouville Eigenvalue Problems
Vritionl Techniques for SturmLiouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment
More informationMath 270A: Numerical Linear Algebra
Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner
More information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More informationA recursive construction of efficiently decodable listdisjunct matrices
CSE 709: Compressed Sensing nd Group Testing. Prt I Lecturers: Hung Q. Ngo nd Atri Rudr SUNY t Bufflo, Fll 2011 Lst updte: October 13, 2011 A recursive construction of efficiently decodble listdisjunct
More informationPositive Solutions of Operator Equations on HalfLine
Int. Journl of Mth. Anlysis, Vol. 3, 29, no. 5, 21122 Positive Solutions of Opertor Equtions on HlfLine Bohe Wng 1 School of Mthemtics Shndong Administrtion Institute Jinn, 2514, P.R. Chin sdusuh@163.com
More informationMultidimensional. MOD Planes. W. B. Vasantha Kandasamy Ilanthenral K Florentin Smarandache
Multidimensionl MOD Plnes W. B. Vsnth Kndsmy Ilnthenrl K Florentin Smrndche 2015 This book cn be ordered from: EuropNov ASBL Clos du Prnsse, 3E 1000, Bruxelles Belgium Emil: info@europnov.be URL: http://www.europnov.be/
More informationON A CONVEXITY PROPERTY. 1. Introduction Most general class of convex functions is defined by the inequality
Krgujevc Journl of Mthemtics Volume 40( (016, Pges 166 171. ON A CONVEXITY PROPERTY SLAVKO SIMIĆ Abstrct. In this rticle we proved n interesting property of the clss of continuous convex functions. This
More informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationSimple Gamma Rings With Involutions.
IOSR Journl of Mthemtics (IOSRJM) ISSN: 22785728. Volume 4, Issue (Nov.  Dec. 2012), PP 4048 Simple Gmm Rings With Involutions. 1 A.C. Pul nd 2 Md. Sbur Uddin 1 Deprtment of Mthemtics University of
More informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 200910 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
More informationTheory of the Integral
Spring 2012 Theory of the Integrl Author: Todd Gugler Professor: Dr. Drgomir Sric My 10, 2012 2 Contents 1 Introduction 5 1.0.1 Office Hours nd Contct Informtion..................... 5 1.1 Set Theory:
More informationMath Solutions to homework 1
Mth 75  Solutions to homework Cédric De Groote October 5, 07 Problem, prt : This problem explores the reltionship between norms nd inner products Let X be rel vector spce ) Suppose tht is norm on X tht
More informationInfinite Geometric Series
Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to
More informationInnerproduct spaces
Innerproduct spces Definition: Let V be rel or complex liner spce over F (here R or C). An inner product is n opertion between two elements of V which results in sclr. It is denoted by u, v nd stisfies:
More informationLecture 19: Continuous Least Squares Approximation
Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for
More informationApproximation of functions belonging to the class L p (ω) β by linear operators
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 3, 9, Approximtion of functions belonging to the clss L p ω) β by liner opertors W lodzimierz Lenski nd Bogdn Szl Abstrct. We prove
More informationIN GAUSSIAN INTEGERS X 3 + Y 3 = Z 3 HAS ONLY TRIVIAL SOLUTIONS A NEW APPROACH
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A2 IN GAUSSIAN INTEGERS X + Y = Z HAS ONLY TRIVIAL SOLUTIONS A NEW APPROACH Elis Lmpkis Lmpropoulou (Term), Kiprissi, T.K: 24500,
More informationLecture Solution of a System of Linear Equation
ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville  D. Keffer, 5/9/98 (updted /) Lecture 8  Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner
More informationM A T H F A L L CORRECTION. Algebra I 2 1 / 0 9 / U N I V E R S I T Y O F T O R O N T O
M A T H 2 4 0 F A L L 2 0 1 4 HOMEWORK ASSIGNMENT #1 CORRECTION Alger I 2 1 / 0 9 / 2 0 1 4 U N I V E R S I T Y O F T O R O N T O 1. Suppose nd re nonzero elements of field F. Using only the field xioms,
More informationPLANAR NORMAL SECTIONS OF FOCAL MANIFOLDS OF ISOPARAMETRIC HYPERSURFACES IN SPHERES
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 56, No. 2, 2015, Pges 119 133 Published online: September 30, 2015 PLANAR NORMAL SECTIONS OF FOCAL MANIFOLDS OF ISOPARAMETRIC HYPERSURFACES IN SPHERES Abstrct.
More informationg i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f
1. Appliction of functionl nlysis to PEs 1.1. Introduction. In this section we give little introduction to prtil differentil equtions. In prticulr we consider the problem u(x) = f(x) x, u(x) = x (1) where
More informationHW3, Math 307. CSUF. Spring 2007.
HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem
More informationAnalytical Methods Exam: Preparatory Exercises
Anlyticl Methods Exm: Preprtory Exercises Question. Wht does it men tht (X, F, µ) is mesure spce? Show tht µ is monotone, tht is: if E F re mesurble sets then µ(e) µ(f). Question. Discuss if ech of the
More informationMath 61CM  Solutions to homework 9
Mth 61CM  Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationMATH 174A: PROBLEM SET 5. Suggested Solution
MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous relvlued function on I), nd let L 1 (I) denote the completion
More informationPOSITIVE SOLUTIONS FOR SINGULAR THREEPOINT BOUNDARYVALUE PROBLEMS
Electronic Journl of Differentil Equtions, Vol. 27(27), No. 156, pp. 1 8. ISSN: 1726691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu (login: ftp) POSITIVE SOLUTIONS
More informationA Note on Heredity for Terraced Matrices 1
Generl Mthemtics Vol. 16, No. 1 (2008), 59 A Note on Heredity for Terrced Mtrices 1 H. Crwford Rhly, Jr. In Memory of Myrt Nylor Rhly (19172006) Abstrct A terrced mtrix M is lower tringulr infinite mtrix
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationSolution to Fredholm Fuzzy Integral Equations with Degenerate Kernel
Int. J. Contemp. Mth. Sciences, Vol. 6, 2011, no. 11, 535543 Solution to Fredholm Fuzzy Integrl Equtions with Degenerte Kernel M. M. Shmivnd, A. Shhsvrn nd S. M. Tri Fculty of Science, Islmic Azd University
More informationS. S. Dragomir. 2, we have the inequality. b a
Bull Koren Mth Soc 005 No pp 3 30 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Abstrct Compnions of Ostrowski s integrl ineulity for bsolutely
More informationUniversity of Houston, Department of Mathematics Numerical Analysis II
University of Houston, Deprtment of Mtemtics Numericl Anlysis II 6 Glerkin metod, finite differences nd colloction 6.1 Glerkin metod Consider sclr 2nd order ordinry differentil eqution in selfdjoint form
More informationPENALIZED LEAST SQUARES FITTING. Manfred von Golitschek and Larry L. Schumaker
Serdic Mth. J. 28 2002), 329348 PENALIZED LEAST SQUARES FITTING Mnfred von Golitschek nd Lrry L. Schumker Communicted by P. P. Petrushev Dedicted to the memory of our collegue Vsil Popov Jnury 4, 942
More informationSOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL (1 + µ(f n )) f(x) =. But we don t need the exact bound.) Set
SOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL 28 Nottion: N {, 2, 3,...}. (Tht is, N.. Let (X, M be mesurble spce with σfinite positive mesure µ. Prove tht there is finite positive mesure ν on (X, M such
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationBoolean Algebra. Boolean Algebras
Boolen Algebr Boolen Algebrs A Boolen lgebr is set B of vlues together with:  two binry opertions, commonly denoted by + nd,  unry opertion, usully denoted by or ~ or,  two elements usully clled zero
More informationWHEN IS A FUNCTION NOT FLAT? 1. Introduction. {e 1 0, x = 0. f(x) =
WHEN IS A FUNCTION NOT FLAT? YIFEI PAN AND MEI WANG Abstrct. In this pper we prove unique continution property for vector vlued functions of one vrible stisfying certin differentil inequlity. Key words:
More informationA HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES. 1. Introduction
Ttr Mt. Mth. Publ. 44 (29), 159 168 DOI: 1.2478/v1127956z t m Mthemticl Publictions A HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES Miloslv Duchoň Peter Mličký ABSTRACT. We present Helly
More informationSturmLiouville Eigenvalue problem: Let p(x) > 0, q(x) 0, r(x) 0 in I = (a, b). Here we assume b > a. Let X C 2 1
Ch.4. INTEGRAL EQUATIONS AND GREEN S FUNCTIONS Ronld B Guenther nd John W Lee, Prtil Differentil Equtions of Mthemticl Physics nd Integrl Equtions. Hildebrnd, Methods of Applied Mthemtics, second edition
More informationAbstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
More informationLecture Note 9: Orthogonal Reduction
MATH : Computtionl Methods of Liner Algebr 1 The Row Echelon Form Lecture Note 9: Orthogonl Reduction Our trget is to solve the norml eution: Xinyi Zeng Deprtment of Mthemticl Sciences, UTEP A t Ax = A
More informationWell Centered Spherical Quadrangles
Beiträge zur Algebr und Geometrie Contributions to Algebr nd Geometry Volume 44 (003), No, 539549 Well Centered Sphericl Qudrngles An M d Azevedo Bred 1 Altino F Sntos Deprtment of Mthemtics, University
More informationVariational problems of some second order Lagrangians given by Pfaff forms
Vritionl problems of some second order Lgrngins given by Pfff forms P. Popescu M. Popescu Abstrct. In this pper we study the dynmics of some second order Lgrngins tht come from Pfff forms i.e. differentil
More informationA PROOF OF THE FUNDAMENTAL THEOREM OF CALCULUS USING HAUSDORFF MEASURES
INROADS Rel Anlysis Exchnge Vol. 26(1), 2000/2001, pp. 381 390 Constntin Volintiru, Deprtment of Mthemtics, University of Buchrest, Buchrest, Romni. emil: cosv@mt.cs.unibuc.ro A PROOF OF THE FUNDAMENTAL
More informationNumerical radius Haagerup norm and square factorization through Hilbert spaces
J Mth Soc Jpn Vol 58, No, 006 Numericl rdius Hgerup norm nd squre fctoriztion through Hilbert spces By Tkshi Itoh nd Msru Ngis (Received Apr 7, 004) (Revised Mr, 005) Abstrct We study fctoriztion of bounded
More informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly welldefined, is too restrictive for mny purposes; there re functions which
More informationMultiple Positive Solutions for the System of Higher Order TwoPoint Boundary Value Problems on Time Scales
Electronic Journl of Qulittive Theory of Differentil Equtions 2009, No. 32, 3; http://www.mth.uszeged.hu/ejqtde/ Multiple Positive Solutions for the System of Higher Order TwoPoint Boundry Vlue Problems
More informationCM10196 Topic 4: Functions and Relations
CM096 Topic 4: Functions nd Reltions Guy McCusker W. Functions nd reltions Perhps the most widely used notion in ll of mthemtics is tht of function. Informlly, function is n opertion which tkes n input
More information