c 2006 Society for Industrial and Applied Mathematics
|
|
- Brittney Rose
- 5 years ago
- Views:
Transcription
1 SIAM J. MATRIX ANAL. APPL. Vol. 7, No. 3, pp c 006 Society for Idustrial ad Applied Mathematics EXTREMAL EIGENVALUES OF REAL SYMMETRIC MATRICES WITH ENTRIES IN AN INTERVAL XINGZHI ZHAN Abstract. We determie the exact rage of the smallest ad largest eigevalues of real symmetric matrices of a give order whose etries are i a give iterval. The maximizig ad miimizig matrices are specified. We also cosider the maximal spread of such matrices. Key words. eigevalue, symmetric matrix, spread AMS subject classificatios. 15A18, 15A4, 15A57 DOI / Itroductio. Let S a, b deote the set of real symmetric matrices whose etries are i the iterval a, b. For a real symmetric matrix A, we always deote the eigevalues of A i decreasig order by λ 1 A) λ A). We will study the smallest eigevalue λ A) ad the largest eigevalue λ 1 A) whe A varies i S a, b. Costatie proved that if A S 0,b, the { b/ if is eve, λ A) 1b/ if is odd. So the matrices treated there are oegative. The proof techiques of are graphtheoretic. I 7 Roth gave aother proof of this result by aalysis of eigevectors. I this paper we will determie the smallest ad largest values of both λ A) ad λ 1 A) whe A S a, b for geeric a<b, thus geeralizig Costatie s result. The spread of a real symmetric matrix A is defied as sa) =λ 1 A) λ A). This quatity has applicatios i combiatorial optimizatio problems 3. Some lower bouds o the spread of Hermitia matrices are kow; see 5 ad the refereces therei. We will determie the maximal value of sa) for A S a, a. We always regard real vectors i R as 1 matrices. A basic fact see 1 or 4) for a real symmetric matrix A is λ A) = mi{x T Ax : x =1,x R }, λ 1 A) = max{x T Ax : x =1,x R }.. Extremal eigevalues. We first cosider the lower boud for the smallest eigevalue. Deote by J r,s the r s matrix with all etries equal to 1, ad write J r for J r,r. Theorem 1. Let A S a, b with ad a<b. i) If a <b, the { a b)/ if is eve, λ A) a ) a + 1)b / if is odd. Received by the editors March 7, 005; accepted for publicatio i revised form) by R. Bhatia July 6, 005; published electroically Jauary 7, 006. This work was supported by NSFC grat Departmet of Mathematics, East Chia Normal Uiversity, Shaghai 0006, Chia zha@ math.ecu.edu.c). 851
2 85 XINGZHI ZHAN If is eve, equality holds if ad oly if A is permutatio similar to a b J b a. If is odd, equality holds if ad oly if A is permutatio similar to aj 1 bj 1, +1. bj +1, 1 aj +1 ii) If a b, the λ A) a. If a >b, equality holds if ad oly if A = aj. If a = b, equality holds if ad oly if A is permutatio similar to aj k bj k, k bj k,k aj k for some k with 1 k. Proof. For ay fixed A S a, b, let x =x 1,...,x ) T be a uit i.e., x =1) eigevector correspodig to λ A). By simultaeous permutatios of the rows ad colums of A if ecessary, we may suppose that x i 0 for i =1,...,k ad x j < 0 for j = k +1,...,,1 k. We eed ot cosider the case k = 0, as i that case we use x istead of x. Let e R be the vector with all etries equal to 1. Deote by A B the Hadamard product of A ad B, i.e., the etrywise product. The we may write x T Ax i a more visible form: 1) λ A) =x T Ax = e T A xx T )e. Note that the matrix xx T is divided ito four blocks: The etries i xx T )1,...,k ad i xx T )k +1,..., are oegative, while the etries i xx T )1,...,k k +1,..., ad i xx T )k +1,..., 1,...,k are opositive. Thus from 1) we see clearly that if we defie aj à = Jk; a, b) k bj ) k, k, the bj k,k aj k λ Ã) = mi{yt Ãy : y =1,y R } x T Ãx x T Ax = λ A). Therefore the smallest value of λ A) for A S a, b ca be attaied at some matrix of the form i ). Sice the rak of Jk; a, b) is at most, it has at most two ozero eigevalues. By cosiderig the trace ad the Frobeius orm we deduce that λ Jk; a, b)) = a ) 3) k) a +4k k)b /. i) Sice a <b,if is eve, the right side of 3) attais its miimum at k = / ad λ A) λ J/; a, b)) = a b)/ for ay A S a, b. If is odd, the right side of 3) attais its miimum at k = 1)/ ad k = +1)/. Hece λ A) λ J 1)/; a, b)) = a ) a + 1)b /.
3 EIGENVALUES OF REAL SYMMETRIC MATRICES 853 Now we prove the equality coditios. First suppose is eve. Let A =a ij ) S a, b such that λ A) =a b)/ ad let x =x 1,...,x ) T be a correspodig uit eigevector. Suppose x has exactly t ozero compoets with t <. By the above bouds, if t is eve, the ad if t is odd, the λ A) =x T Ax λ A) =x T Ax ta b)/ >a b)/, ta a b )+t b ) / >ta b)/ >a b)/, both cotradictig the assumptio that λ A) =a b)/. Therefore all the compoets of x are ozero. Suppose x has k positive compoets ad k egative compoets. From 3) we kow that if k /, λ A) λ Jk; a, b)) >λ J/; a, b)) = a b)/, a cotradictio. Thus we must have k = /. By simultaeous row ad colum permutatios of A if ecessary, we may suppose that x i > 0 for i =1,...,/ ad x j < 0 for j =/)+1,...,. The λ A) =x T Ax x T J/; a, b)x λ A) forces A = J/; a, b), sice otherwise the first iequality above will be strict, which is impossible. Therefore the origial A is permutatio similar to J/; a, b). The equality coditio for the case whe is odd ca be similarly proved. Just ote that ta b) >a a + 1)b for 1 t 1, the lower boud a a + 1)b )/ is strictly decreasig i, ad aj +1 bj +1, 1 aj 1 bj 1 ad, +1 bj 1, +1 aj 1 bj +1, 1 aj +1 are permutatio similar. ii) a b ad a<bimply a<0. If a >b, the miimum of the right side of 3) is attaied at k =, ad if a = b, the right side of 3) has equal values for all k with 1 k. I ay case the miimum is a. This proves λ A) a. The proof of the equality coditios is similar to that of case i) ad we omit the details. Sice A S a, b implies A aj S 0,b a, it is atural to ask whether Theorem 1 ca be deduced from Costatie s result by usig the perturbatio iequality: λ G + H) λ G)+λ H) for real symmetric matrices G, H 1. I geeral the aswer is o. Let us examie the case 0 <a<b. If is odd, the perturbatio iequality ad Costatie s result give λ A) =λ A aj )+aj λ A aj )+λ aj ) = λ A aj ) 1a b)/.
4 854 XINGZHI ZHAN It is easy to verify that this lower boud 1a b)/ is strictly less tha the sharp boud a a + 1)b )/ i Theorem 1. O the other had, if is eve, the lower boud a b)/ ca ideed be deduced from Costatie s result. For a real symmetric matrix A, λ 1 A) = λ A). Also a a ij b is equivalet to b a ij a. Thus the followig corollary o upper bouds for the largest eigevalue follows from Theorem 1. Corollary. Let A S a, b with ad a<b. i) If a< b, the { b a)/ if is eve, λ 1 A) b + ) b + 1)a / if is odd. If is eve, equality holds if ad oly if A is permutatio similar to b a J a b. If is odd, equality holds if ad oly if A is permutatio similar to bj 1 aj 1, +1. aj +1, 1 bj +1 ii) If a b, the λ 1 A) b. If a> b, equality holds if ad oly if A = bj.ifa = b, equality holds if ad oly if A is permutatio similar to bj k aj k, k aj k,k bj k for some k with 1 k. Now we tur to the study of upper bouds o the smallest eigevalue ad lower bouds o the largest eigevalue. For real matrices A, B, we write A B to mea that B A is etrywise oegative. We eed the followig two lemmas. Lemma 3 see 1 or 4). Let H be a real symmetric matrix of order ad G be a pricipal submatrix of order k of H. The λ j H) λ j G) λ j+ k H) for j =1,...,k. Lemma 4 see 6, p. 38). Let A, B be oegative matrices of the same order satisfyig A B. The ρa) ρb), where ρ ) is the Perro root spectral radius). If, i additio, A B ad B is irreducible, the ρa) <ρb). Sice λ A) = λ 1 A) ad λ 1 A) = λ A) for real symmetric matrices A, for our problem there are essetially two differet cases: 0 <a<bad a 0 <b. Deote by J ad I the all-oe matrix ad the idetity matrix, respectively. Theorem 5. Let A S a, b with ad a<b. i) Let 0 <a<b. The 4) λ A) b a. Equality i 4) holds if ad oly if A = aj +b a)i. 5) λ 1 A) a. Equality i 5) holds if ad oly if A = aj.
5 ii) Let a 0 <b. The EIGENVALUES OF REAL SYMMETRIC MATRICES 855 6) λ A) b. Equality i 6) holds if ad oly if A = bi. 7) λ 1 A) a. Equality i 7) holds if ad oly if A = ai. Proof. i) Let A =a ij ). For i<j, by Lemma 3 we have 8) λ A) λ aii a ji a ij a jj ) = a ii + a jj a ii a jj ) +4a ij a ii + a jj a ij b a. Thus λ A) b a. If λ A) =b a, all the iequalities i 8) must be equality. This forces a ii = a jj = b ad a ij = a ji = a. As this should be true for all i<j, A = aj +b a)i. A S a, b implies A aj 0. By Lemma 4 ad the Perro Frobeius theory 6, λ 1 A) =ρa) ρaj) =λ 1 aj) =a. If A S a, b ad A aj, the sice A is irreducible A is i fact etrywise positive), agai by Lemma 4, λ 1 A) >λ 1 aj) =a. Thusλ 1 A) =a if ad oly if A = aj. ii) λ A) tra b = b. If λ A) =b, the tra = b ad cosequetly a ii = b, i =1,...,. For ay i<j,by Lemma 3 we have b aij b = λ A) λ a ji b Thus a ij = 0 for all i<j, i.e., A = bi. λ 1 A) tra ) = b a ij. a = a. If λ 1 A) =a, the tra = a ad hece a ii = a, i =1,...,. For ay i<j,by Lemma 3 we have a aij a = λ 1 A) λ 1 a ji a ) = a + a ij. So a ij = 0 for all i<j, i.e., A = ai. This completes the proof.
6 856 XINGZHI ZHAN 3. The maximal spread. Deote by sa) the spread of A. We treat oly the case whe the iterval is symmetric about the origi. Of course we may use the upper boud o λ 1 i Corollary ad the lower boud o λ i Theorem 1 to give a upper boud o the spread λ 1 λ, but that boud is ot sharp. This is because the upper boud o λ 1 ad the lower boud o λ caot be simultaeously attaied at oe commo matrix. A {±1}-matrix is a matrix whose etries are either 1 or 1. Two matrices A, B of the same order are said to be D-similar if there is a diagoal matrix D with diagoal etries equal to 1 or 1 such that DAD = B. We will eed the followig lemma. Lemma 6. Let A be a symmetric {±1}-matrix with all diagoal etries equal to 1. The either A is D-similar to J or A has a pricipal submatrix of order 3 which is similar to B 1 = Proof. The cases =1, are obvious. Use iductio o for 3. First let =3. IfA = J 3, there is othig to prove. Otherwise A has a off-diagoal etry equal to 1. The there are the followig possibilities of A: B 1,, 1 1 1, 1 1 1, 1 1 1, , It is easy to check that each of these matrices satisfies the coclusio. Now cosider 4 ad suppose the lemma holds for matrices of order 1. Let 1 u T A =. u A 1 If A 1 has a pricipal submatrix of order 3 which is similar to B 1, the so does A. Otherwise, by the assumptio, A 1 is D-similar to J 1.SoAis D-similar to 1 v T H = =h v J ij ). 1 If each etry of v is 1 or each etry of v is 1, the H ad hece A are D-similar to J. Otherwise we have h 1,p = 1, h 1,q =1orh 1,p =1,h 1,q = 1 for some 1 <p<q. I the first case H1,p,q=B 1. I the secod case H1,p,q is D-similar to B 1. Therefore i both cases A has a pricipal submatrix which is similar to B 1. Two matrices A, B of the same order are said to be sig-permutatio similar if there exist a permutatio matrix P ad a diagoal matrix D with diagoal etries equal to 1 or 1 such that DP T AP D = B. It is clear that sig-permutatio similarity is a equivalece relatio. Theorem 7. Let A S a, a with ad a>0. The { a if is eve, sa) 1a if is odd.
7 EIGENVALUES OF REAL SYMMETRIC MATRICES 857 If is eve, equality holds if ad oly if A is sig-permutatio similar to 1 1 a J 1 1. If is odd, equality holds if ad oly if A is sig-permutatio similar to J +1 J +1 ±a, 1. J 1, +1 J 1 Proof. By cosiderig a 1 A istead of A, it suffices to prove the theorem for the case a = 1. Suppose A S 1, 1. Throughout this proof we write λ j for λ j A). For x R, we always write its compoets as x 1,...,x. Give A S 1, 1, let x, y R be uit eigevectors such that λ 1 = x T Ax, λ = y T Ay. The 9) sa) =x T Ax y T Ay = e T A xx T yy T )e. Note that the i, j) etry of xx T yy T is x i x j y i y j. We defie a ew matrix à =ã ij) as ã ij =1ifx i x j y i y j 0 ad ã ij = 1 ifx i x j y i y j < 0. The from 9) we have sa) x T Ãx y T Ãy max{w T Ãw : w =1,w R } mi{z T Ãz : z =1,z R } = sã). Therefore the maximal spread ca always be attaied at some {±1}-matrix. If =, the coclusios of the theorem are easily checked to be true. Next we assume 3. Now suppose A is a {±1}-matrix of order. The followig three matrices will play a role i our proof: B 1 = Their eigevalues are, B = 1 1 1, B 3 = λb 1 )={,, 1}, λb )={, 1, }, λb 3 )={, 1, }. If A has a pricipal submatrix of order 3 which is similar to B 1, the by Lemma 3 λ. If A has a pricipal submatrix of order 3 which is similar to B, the by Lemma 3 λ 1. If A has a pricipal submatrix of order 3 which is similar to B 3, the by Lemma 3 λ 1 1. I all three cases we have 1 λ 1 + λ = A F λ j 1. j=. Hece 10) sa) =λ 1 λ λ 1 + λ ) { } 1) < mi, 1. Thus if oe of the above cases occurs, the spread is less tha our claimed upper boud.
8 858 XINGZHI ZHAN If all the diagoal etries of A are 1, the by Lemma 6 either A is D-similar to J or A has a pricipal submatrix of order 3 which is similar to B 1. Sice sj )=, i both cases sa) is ot the maximal value. If all the diagoal etries of A are 1, the sice s A) =sa), this case is the same as what we just discussed. Next cosider those {±1}-matrices A whose diagoal cotais both 1 ad 1. By simultaeous row ad colum permutatios if ecessary, we may suppose Ar A A = r,s A T, r,s A s where A r is of order r 1 r 1) ad A r s diagoal etries are all 1, ad A s is of order s = r ad A s s diagoal etries are all 1. Sice s A) =sa), we eed cosider oly the case r s. The r. By Lemma 6, either A r is D-similar to J r or A r has a pricipal submatrix of order 3 which is similar to B 1. I the first case A is D-similar to a matrix whose rth leadig pricipal submatrix is J r, while i the secod case sa) is ot the maximal value. Thus we may suppose A r = J r. Now two cases ca occur. i) A r,s has a colum which cotais both 1 ad 1. The A has a pricipal submatrix which is similar to B, ad sa) is ot the maximal value. ii) Each colum of A r,s cotais oly 1 or oly 1. The A is D-similar to Jr J r,s G =, J s,r where all the diagoal etries of Ãs remai 1. If Ãs has a off-diagoal etry equal to 1, the G has B 3 as a pricipal submatrix ad sg) is ot the maximal value. Thus we further cosider the case Ãs = J s. Now there remais the case Jr J 11) A = r,s, J s,r J s à s where r s. It is easy to see that Jr J 1) s r,s J s,r J s ) = r ). Therefore by 1) the maximal spread of the matrices i 11) is attaied uiquely at r = / if is eve, ad the maximal spread is 1 attaied uiquely at r = +1)/ if is odd. For the odd case ote that we have assumed r s = r, ad hece r = 1)/ does ot occur. At this stage we have foud the maximal spread of A S 1, 1. Next we determie those matrices which attai the maximal spread. Suppose A S 1, 1 attais the maximal spread ad x, y are the uit eigevectors correspodig to λ 1 ad λ, respectively. Assume that xx T yy T has a zero etry, say, x i x j y i y j =0 for some i, j. From 9) it is clear that we ca chage the correspodig etry a ij of A arbitrarily without affectig the value of sa). So we may suppose a ij = 0. The 1 A F λ 1 + λ λ 1 λ ) { if is eve, = 1 if is odd, which is a cotradictio. Thus every etry of xx T yy T is ozero. By 9) we deduce that A must be a {±1}-matrix. O the other had, the above aalysis leadig
9 EIGENVALUES OF REAL SYMMETRIC MATRICES 859 to the maximal spread shows that whe is eve, sa) = if ad oly if A is sig-permutatio similar to 1 1 J 1 1, ad whe is odd, sa) = 1 if ad oly if A is sig-permutatio similar to J +1 J +1 ±, 1. J 1, +1 J 1 The permutatio similarity comes from our operatio to put positive diagoal etries together ad let them appear first. The possible mius sig comes from the fact that s A) = sa), which we used to simplify our aalysis. Note also that i the case whe is eve, J 1 1 ad 1 1 J are sig-permutatio similar, ad the mius sig eed ot appear i our assertio for the equality case. This completes the proof. Let e t R t deote the all-oe vector. By Theorem 7 ad 9) we get the followig iterestig corollary. Corollary 8. max x i x j y i y j : x = y =1,x,y R = i,j=1 { if is eve, 1 if is odd. The maximum is attaied at x, y if ad oly if x = DPx 0, y = DPy 0 for some diagoal matrix D with diagoal etries equal to 1 or 1 ad some permutatio matrix P where ) ) ae/ be/ x 0 =, y be 0 =, a =1+ 1 )b, b = / ae / + ) if is eve ad x 0 = y 0 = ae+1)/ be 1)/ ce+1)/ de 1)/ ),a= ),c= ),b= 1 ),d= ) 1 ) 1 if is odd. We remark that the above x 0 ad y 0 are the uit eigevectors correspodig to the largest ad smallest eigevalues of the maximizig matrix i Theorem 7. We have the followig two obvious problems which are ot solved here. Problem 1. For a give iteger j with j 1, determie max{λ j A) :A S a, b}, mi{λ j A) :A S a, b}
10 860 XINGZHI ZHAN ad determie which matrices attai the maximum ad which matrices attai the miimum. Problem. For geeric a<b, determie max{sa) :A S a, b}, where sa) deotes the spread of A, ad determie which matrices attai the maximum. REFERENCES 1 R. Bhatia, Matrix Aalysis, Spriger-Verlag, New York, G. Costatie, Lower bouds o the spectra of symmetric matrices with oegative etries, Liear Algebra Appl., ), pp G. Fike, R. E. Burkard, ad F. Redl, Quadratic assigmet problems, i Surveys of Combiatorial Optimizatio, North Hollad, Amsterdam, 1987, pp R. A. Hor ad C. R. Johso, Matrix Aalysis, Cambridge Uiversity Press, Cambridge, UK, E. Jiag ad X. Zha, Lower bouds for the spread of a Hermitia matrix, Liear Algebra Appl., ), pp H. Mic, Noegative Matrices, Joh Wiley & Sos, New York, R. Roth, O the eigevectors belogig to the miimum eigevalue of a essetially oegative symmetric matrix with bipartite graph, Liear Algebra Appl., ), pp
The inverse eigenvalue problem for symmetric doubly stochastic matrices
Liear Algebra ad its Applicatios 379 (004) 77 83 www.elsevier.com/locate/laa The iverse eigevalue problem for symmetric doubly stochastic matrices Suk-Geu Hwag a,,, Sug-Soo Pyo b, a Departmet of Mathematics
More informationA Hadamard-type lower bound for symmetric diagonally dominant positive matrices
A Hadamard-type lower boud for symmetric diagoally domiat positive matrices Christopher J. Hillar, Adre Wibisoo Uiversity of Califoria, Berkeley Jauary 7, 205 Abstract We prove a ew lower-boud form of
More informationOn Nonsingularity of Saddle Point Matrices. with Vectors of Ones
Iteratioal Joural of Algebra, Vol. 2, 2008, o. 4, 197-204 O Nosigularity of Saddle Poit Matrices with Vectors of Oes Tadeusz Ostrowski Istitute of Maagemet The State Vocatioal Uiversity -400 Gorzów, Polad
More information1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
More informationMatrix Theory, Math6304 Lecture Notes from October 25, 2012 taken by Manisha Bhardwaj
Matrix Theory, Math6304 Lecture Notes from October 25, 2012 take by Maisha Bhardwaj Last Time (10/23/12) Example for low-rak perturbatio, re-examied Relatig eigevalues of matrices ad pricipal submatrices
More informationMath 778S Spectral Graph Theory Handout #3: Eigenvalues of Adjacency Matrix
Math 778S Spectral Graph Theory Hadout #3: Eigevalues of Adjacecy Matrix The Cartesia product (deoted by G H) of two simple graphs G ad H has the vertex-set V (G) V (H). For ay u, v V (G) ad x, y V (H),
More informationStochastic Matrices in a Finite Field
Stochastic Matrices i a Fiite Field Abstract: I this project we will explore the properties of stochastic matrices i both the real ad the fiite fields. We first explore what properties 2 2 stochastic matrices
More informationLecture 8: October 20, Applications of SVD: least squares approximation
Mathematical Toolkit Autum 2016 Lecturer: Madhur Tulsiai Lecture 8: October 20, 2016 1 Applicatios of SVD: least squares approximatio We discuss aother applicatio of sigular value decompositio (SVD) of
More informationBounds for the Extreme Eigenvalues Using the Trace and Determinant
ISSN 746-7659, Eglad, UK Joural of Iformatio ad Computig Sciece Vol 4, No, 9, pp 49-55 Bouds for the Etreme Eigevalues Usig the Trace ad Determiat Qi Zhog, +, Tig-Zhu Huag School of pplied Mathematics,
More informationEigenvalue localization for complex matrices
Electroic Joural of Liear Algebra Volume 7 Article 1070 014 Eigevalue localizatio for complex matrices Ibrahim Halil Gumus Adıyama Uiversity, igumus@adiyama.edu.tr Omar Hirzallah Hashemite Uiversity, o.hirzal@hu.edu.jo
More informationA Note on the Symmetric Powers of the Standard Representation of S n
A Note o the Symmetric Powers of the Stadard Represetatio of S David Savitt 1 Departmet of Mathematics, Harvard Uiversity Cambridge, MA 0138, USA dsavitt@mathharvardedu Richard P Staley Departmet of Mathematics,
More informationTopics in Eigen-analysis
Topics i Eige-aalysis Li Zajiag 28 July 2014 Cotets 1 Termiology... 2 2 Some Basic Properties ad Results... 2 3 Eige-properties of Hermitia Matrices... 5 3.1 Basic Theorems... 5 3.2 Quadratic Forms & Noegative
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More informationMath 510 Assignment 6 Due date: Nov. 26, 2012
1 If A M is Hermitia, show that Math 510 Assigmet 6 Due date: Nov 6, 01 tr A rak (Atr (A with equality if ad oly if A = ap for some a R ad orthogoal projectio P M (ie P = P = P Solutio: Let A = A If A
More informationA new error bound for linear complementarity problems for B-matrices
Electroic Joural of Liear Algebra Volume 3 Volume 3: (206) Article 33 206 A ew error boud for liear complemetarity problems for B-matrices Chaoqia Li Yua Uiversity, lichaoqia@yueduc Megtig Ga Shaorog Yag
More informationEigenvalues and Eigenvectors
5 Eigevalues ad Eigevectors 5.3 DIAGONALIZATION DIAGONALIZATION Example 1: Let. Fid a formula for A k, give that P 1 1 = 1 2 ad, where Solutio: The stadard formula for the iverse of a 2 2 matrix yields
More information1 Last time: similar and diagonalizable matrices
Last time: similar ad diagoalizable matrices Let be a positive iteger Suppose A is a matrix, v R, ad λ R Recall that v a eigevector for A with eigevalue λ if v ad Av λv, or equivaletly if v is a ozero
More informationA 2nTH ORDER LINEAR DIFFERENCE EQUATION
A 2TH ORDER LINEAR DIFFERENCE EQUATION Doug Aderso Departmet of Mathematics ad Computer Sciece, Cocordia College Moorhead, MN 56562, USA ABSTRACT: We give a formulatio of geeralized zeros ad (, )-discojugacy
More informationMATH10212 Linear Algebra B Proof Problems
MATH22 Liear Algebra Proof Problems 5 Jue 26 Each problem requests a proof of a simple statemet Problems placed lower i the list may use the results of previous oes Matrices ermiats If a b R the matrix
More informationA class of spectral bounds for Max k-cut
A class of spectral bouds for Max k-cut Miguel F. Ajos, José Neto December 07 Abstract Let G be a udirected ad edge-weighted simple graph. I this paper we itroduce a class of bouds for the maximum k-cut
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationCHAPTER 10. Majorization and Matrix Inequalities. 10.1/1 Find two vectors x, y R 3 such that neither x y nor x y holds.
CHAPTER 10 Majorizatio ad Matrix Iequalities 10.1/1 Fid two vectors x, y R 3 such that either x y or x y holds. Solutio: Take x = (1, 1 ad y = (2, 3. 10.1/2 Let y = (2, 1 R 2. Sketch the followig sets
More informationMA131 - Analysis 1. Workbook 3 Sequences II
MA3 - Aalysis Workbook 3 Sequeces II Autum 2004 Cotets 2.8 Coverget Sequeces........................ 2.9 Algebra of Limits......................... 2 2.0 Further Useful Results........................
More informationPROBLEM SET I (Suggested Solutions)
Eco3-Fall3 PROBLE SET I (Suggested Solutios). a) Cosider the followig: x x = x The quadratic form = T x x is the required oe i matrix form. Similarly, for the followig parts: x 5 b) x = = x c) x x x x
More informationCSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationA LIMITED ARITHMETIC ON SIMPLE CONTINUED FRACTIONS - II 1. INTRODUCTION
A LIMITED ARITHMETIC ON SIMPLE CONTINUED FRACTIONS - II C. T. LONG J. H. JORDAN* Washigto State Uiversity, Pullma, Washigto 1. INTRODUCTION I the first paper [2 ] i this series, we developed certai properties
More informationSpectral Partitioning in the Planted Partition Model
Spectral Graph Theory Lecture 21 Spectral Partitioig i the Plated Partitio Model Daiel A. Spielma November 11, 2009 21.1 Itroductio I this lecture, we will perform a crude aalysis of the performace of
More informationMath 61CM - Solutions to homework 3
Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More informationON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS
ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS NORBERT KAIBLINGER Abstract. Results of Lid o Lehmer s problem iclude the value of the Lehmer costat of the fiite cyclic group Z/Z, for 5 ad all odd. By complemetary
More informationPAijpam.eu ON TENSOR PRODUCT DECOMPOSITION
Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314
More informationGeneralization of Samuelson s inequality and location of eigenvalues
Proc. Idia Acad. Sci. Math. Sci.) Vol. 5, No., February 05, pp. 03. c Idia Academy of Scieces Geeralizatio of Samuelso s iequality ad locatio of eigevalues R SHARMA ad R SAINI Departmet of Mathematics,
More informationTHE SPECTRAL RADII AND NORMS OF LARGE DIMENSIONAL NON-CENTRAL RANDOM MATRICES
COMMUN. STATIST.-STOCHASTIC MODELS, 0(3), 525-532 (994) THE SPECTRAL RADII AND NORMS OF LARGE DIMENSIONAL NON-CENTRAL RANDOM MATRICES Jack W. Silverstei Departmet of Mathematics, Box 8205 North Carolia
More informationOn the distribution of coefficients of powers of positive polynomials
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 49 (2011), Pages 239 243 O the distributio of coefficiets of powers of positive polyomials László Major Istitute of Mathematics Tampere Uiversity of Techology
More informationOn Some Properties of Digital Roots
Advaces i Pure Mathematics, 04, 4, 95-30 Published Olie Jue 04 i SciRes. http://www.scirp.org/joural/apm http://dx.doi.org/0.436/apm.04.46039 O Some Properties of Digital Roots Ilha M. Izmirli Departmet
More informationA NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS
Acta Math. Hugar., 2007 DOI: 10.1007/s10474-007-7013-6 A NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS L. L. STACHÓ ad L. I. SZABÓ Bolyai Istitute, Uiversity of Szeged, Aradi vértaúk tere 1, H-6720
More informationBrief Review of Functions of Several Variables
Brief Review of Fuctios of Several Variables Differetiatio Differetiatio Recall, a fuctio f : R R is differetiable at x R if ( ) ( ) lim f x f x 0 exists df ( x) Whe this limit exists we call it or f(
More informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationYuki Seo. Received May 23, 2010; revised August 15, 2010
Scietiae Mathematicae Japoicae Olie, e-00, 4 45 4 A GENERALIZED PÓLYA-SZEGÖ INEQUALITY FOR THE HADAMARD PRODUCT Yuki Seo Received May 3, 00; revised August 5, 00 Abstract. I this paper, we show a geeralized
More informationEnd-of-Year Contest. ERHS Math Club. May 5, 2009
Ed-of-Year Cotest ERHS Math Club May 5, 009 Problem 1: There are 9 cois. Oe is fake ad weighs a little less tha the others. Fid the fake coi by weighigs. Solutio: Separate the 9 cois ito 3 groups (A, B,
More informationw (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.
2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of o-time jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For
More informationFastest mixing Markov chain on a path
Fastest mixig Markov chai o a path Stephe Boyd Persi Diacois Ju Su Li Xiao Revised July 2004 Abstract We ider the problem of assigig trasitio probabilities to the edges of a path, so the resultig Markov
More informationDIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS
DIVISIBILITY PROPERTIES OF GENERALIZED FIBONACCI POLYNOMIALS VERNER E. HOGGATT, JR. Sa Jose State Uiversity, Sa Jose, Califoria 95192 ad CALVIN T. LONG Washigto State Uiversity, Pullma, Washigto 99163
More informationComputation of Error Bounds for P-matrix Linear Complementarity Problems
Mathematical Programmig mauscript No. (will be iserted by the editor) Xiaoju Che Shuhuag Xiag Computatio of Error Bouds for P-matrix Liear Complemetarity Problems Received: date / Accepted: date Abstract
More informationNotes for Lecture 11
U.C. Berkeley CS78: Computatioal Complexity Hadout N Professor Luca Trevisa 3/4/008 Notes for Lecture Eigevalues, Expasio, ad Radom Walks As usual by ow, let G = (V, E) be a udirected d-regular graph with
More informationMDIV. Multiple divisor functions
MDIV. Multiple divisor fuctios The fuctios τ k For k, defie τ k ( to be the umber of (ordered factorisatios of ito k factors, i other words, the umber of ordered k-tuples (j, j 2,..., j k with j j 2...
More informationThe Perturbation Bound for the Perron Vector of a Transition Probability Tensor
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Liear Algebra Appl. ; : 6 Published olie i Wiley IterSciece www.itersciece.wiley.com. DOI:./la The Perturbatio Boud for the Perro Vector of a Trasitio
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationSingular value decomposition. Mathématiques appliquées (MATH0504-1) B. Dewals, Ch. Geuzaine
Lecture 11 Sigular value decompositio Mathématiques appliquées (MATH0504-1) B. Dewals, Ch. Geuzaie V1.2 07/12/2018 1 Sigular value decompositio (SVD) at a glace Motivatio: the image of the uit sphere S
More informationCONSTRUCTIONS OF TRACE ZERO SYMMETRIC STOCHASTIC MATRICES FOR THE INVERSE EIGENVALUE PROBLEM ROBERT REAMS
CONSTRUCTIONS OF TRACE ZERO SYMMETRIC STOCHASTIC MATRICES FOR THE INVERSE EIGENVALUE PROBLEM ROBERT REAMS Abstract. I the special case of where the spectrum σ = {λ 1,λ 2,λ, 0, 0,...,0} has at most three
More informationTheorem: Let A n n. In this case that A does reduce to I, we search for A 1 as the solution matrix X to the matrix equation A X = I i.e.
Theorem: Let A be a square matrix The A has a iverse matrix if ad oly if its reduced row echelo form is the idetity I this case the algorithm illustrated o the previous page will always yield the iverse
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationRandom Matrices with Blocks of Intermediate Scale Strongly Correlated Band Matrices
Radom Matrices with Blocks of Itermediate Scale Strogly Correlated Bad Matrices Jiayi Tog Advisor: Dr. Todd Kemp May 30, 07 Departmet of Mathematics Uiversity of Califoria, Sa Diego Cotets Itroductio Notatio
More informationMA131 - Analysis 1. Workbook 2 Sequences I
MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................
More informationInverse Matrix. A meaning that matrix B is an inverse of matrix A.
Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix
More informationChapter 2. Periodic points of toral. automorphisms. 2.1 General introduction
Chapter 2 Periodic poits of toral automorphisms 2.1 Geeral itroductio The automorphisms of the two-dimesioal torus are rich mathematical objects possessig iterestig geometric, algebraic, topological ad
More informationResearch Article Some E-J Generalized Hausdorff Matrices Not of Type M
Abstract ad Applied Aalysis Volume 2011, Article ID 527360, 5 pages doi:10.1155/2011/527360 Research Article Some E-J Geeralized Hausdorff Matrices Not of Type M T. Selmaogullari, 1 E. Savaş, 2 ad B. E.
More informationLecture 23 Rearrangement Inequality
Lecture 23 Rearragemet Iequality Holde Lee 6/4/ The Iequalities We start with a example Suppose there are four boxes cotaiig $0, $20, $50 ad $00 bills, respectively You may take 2 bills from oe box, 3
More informationSome Results on Certain Symmetric Circulant Matrices
Joural of Iformatics ad Mathematical Scieces Vol 7, No, pp 81 86, 015 ISSN 0975-5748 olie; 0974-875X prit Pulished y RGN Pulicatios http://wwwrgpulicatioscom Some Results o Certai Symmetric Circulat Matrices
More informationCMSE 820: Math. Foundations of Data Sci.
Lecture 17 8.4 Weighted path graphs Take from [10, Lecture 3] As alluded to at the ed of the previous sectio, we ow aalyze weighted path graphs. To that ed, we prove the followig: Theorem 6 (Fiedler).
More informationCHAPTER I: Vector Spaces
CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig
More informationSequences and Limits
Chapter Sequeces ad Limits Let { a } be a sequece of real or complex umbers A ecessary ad sufficiet coditio for the sequece to coverge is that for ay ɛ > 0 there exists a iteger N > 0 such that a p a q
More informationAn analog of the arithmetic triangle obtained by replacing the products by the least common multiples
arxiv:10021383v2 [mathnt] 9 Feb 2010 A aalog of the arithmetic triagle obtaied by replacig the products by the least commo multiples Bair FARHI bairfarhi@gmailcom MSC: 11A05 Keywords: Al-Karaji s triagle;
More informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
More informationSimple Polygons of Maximum Perimeter Contained in a Unit Disk
Discrete Comput Geom (009) 1: 08 15 DOI 10.1007/s005-008-9093-7 Simple Polygos of Maximum Perimeter Cotaied i a Uit Disk Charles Audet Pierre Hase Frédéric Messie Received: 18 September 007 / Revised:
More informationLinear Algebra and its Applications
Liear Algebra ad its Applicatios 433 (2010) 1148 1153 Cotets lists available at ScieceDirect Liear Algebra ad its Applicatios joural homepage: www.elsevier.com/locate/laa The algebraic coectivity of graphs
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationarxiv: v1 [math.fa] 3 Apr 2016
Aticommutator Norm Formula for Proectio Operators arxiv:164.699v1 math.fa] 3 Apr 16 Sam Walters Uiversity of Norther British Columbia ABSTRACT. We prove that for ay two proectio operators f, g o Hilbert
More informationThe minimum value and the L 1 norm of the Dirichlet kernel
The miimum value ad the L orm of the Dirichlet kerel For each positive iteger, defie the fuctio D (θ + ( cos θ + cos θ + + cos θ e iθ + + e iθ + e iθ + e + e iθ + e iθ + + e iθ which we call the (th Dirichlet
More informationTRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES. 1. Introduction
Math Appl 6 2017, 143 150 DOI: 1013164/ma201709 TRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES PANKAJ KUMAR DAS ad LALIT K VASHISHT Abstract We preset some iequality/equality for traces of Hadamard
More informationA NOTE ON SPECTRAL CONTINUITY. In Ho Jeon and In Hyoun Kim
Korea J. Math. 23 (2015), No. 4, pp. 601 605 http://dx.doi.org/10.11568/kjm.2015.23.4.601 A NOTE ON SPECTRAL CONTINUITY I Ho Jeo ad I Hyou Kim Abstract. I the preset ote, provided T L (H ) is biquasitriagular
More information1+x 1 + α+x. x = 2(α x2 ) 1+x
Math 2030 Homework 6 Solutios # [Problem 5] For coveiece we let α lim sup a ad β lim sup b. Without loss of geerality let us assume that α β. If α the by assumptio β < so i this case α + β. By Theorem
More informationComplex Analysis Spring 2001 Homework I Solution
Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle
More informationTHE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS
THE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS DEMETRES CHRISTOFIDES Abstract. Cosider a ivertible matrix over some field. The Gauss-Jorda elimiatio reduces this matrix to the idetity
More informationCompositions of Random Functions on a Finite Set
Compositios of Radom Fuctios o a Fiite Set Aviash Dalal MCS Departmet, Drexel Uiversity Philadelphia, Pa. 904 ADalal@drexel.edu Eric Schmutz Drexel Uiversity ad Swarthmore College Philadelphia, Pa., 904
More informationThe Choquet Integral with Respect to Fuzzy-Valued Set Functions
The Choquet Itegral with Respect to Fuzzy-Valued Set Fuctios Weiwei Zhag Abstract The Choquet itegral with respect to real-valued oadditive set fuctios, such as siged efficiecy measures, has bee used i
More informationLinear chord diagrams with long chords
Liear chord diagrams with log chords Everett Sulliva Departmet of Mathematics Dartmouth College Haover New Hampshire, U.S.A. everett..sulliva@dartmouth.edu Submitted: Feb 7, 2017; Accepted: Oct 7, 2017;
More informationBeurling Integers: Part 2
Beurlig Itegers: Part 2 Isomorphisms Devi Platt July 11, 2015 1 Prime Factorizatio Sequeces I the last article we itroduced the Beurlig geeralized itegers, which ca be represeted as a sequece of real umbers
More informationQUASIRANDOMNESS AND GOWERS THEOREM. August 16, Contents. 1. Lindsey s Lemma: An Illustration of quasirandomness
QUASIRANDOMNESS AND GOWERS THEOREM QIAN ZHANG August 16, 2007 Abstract. Quasiradomess will be described ad the Quasiradomess Theorem will be used to prove Gowers Theorem. This article assumes some familiarity
More informationMachine Learning Theory Tübingen University, WS 2016/2017 Lecture 12
Machie Learig Theory Tübige Uiversity, WS 06/07 Lecture Tolstikhi Ilya Abstract I this lecture we derive risk bouds for kerel methods. We will start by showig that Soft Margi kerel SVM correspods to miimizig
More informationAchieving Stationary Distributions in Markov Chains. Monday, November 17, 2008 Rice University
Istructor: Achievig Statioary Distributios i Markov Chais Moday, November 1, 008 Rice Uiversity Dr. Volka Cevher STAT 1 / ELEC 9: Graphical Models Scribe: Rya E. Guerra, Tahira N. Saleem, Terrace D. Savitsky
More information6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.
6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationDisjoint unions of complete graphs characterized by their Laplacian spectrum
Electroic Joural of Liear Algebra Volume 18 Volume 18 (009) Article 56 009 Disjoit uios of complete graphs characterized by their Laplacia spectrum Romai Boulet boulet@uiv-tlse.fr Follow this ad additioal
More informationWeek 5-6: The Binomial Coefficients
Wee 5-6: The Biomial Coefficiets March 6, 2018 1 Pascal Formula Theorem 11 (Pascal s Formula For itegers ad such that 1, ( ( ( 1 1 + 1 The umbers ( 2 ( 1 2 ( 2 are triagle umbers, that is, The petago umbers
More informationDefinitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients.
Defiitios ad Theorems Remember the scalar form of the liear programmig problem, Miimize, Subject to, f(x) = c i x i a 1i x i = b 1 a mi x i = b m x i 0 i = 1,2,, where x are the decisio variables. c, b,
More informationMatrix Theory, Math6304 Lecture Notes from October 23, 2012 taken by Satish Pandey
Matrix Theory, Math6304 Lecture Notes from October 3, 01 take by Satish Padey Warm up from last time Example of low rak perturbatio; re-examied We had this operator S = 0 1 0......... 1 1 0 M (C) S is
More informationMathematical Induction
Mathematical Iductio Itroductio Mathematical iductio, or just iductio, is a proof techique. Suppose that for every atural umber, P() is a statemet. We wish to show that all statemets P() are true. I a
More information5.1 Review of Singular Value Decomposition (SVD)
MGMT 69000: Topics i High-dimesioal Data Aalysis Falll 06 Lecture 5: Spectral Clusterig: Overview (cotd) ad Aalysis Lecturer: Jiamig Xu Scribe: Adarsh Barik, Taotao He, September 3, 06 Outlie Review of
More informationSquare-Congruence Modulo n
Square-Cogruece Modulo Abstract This paper is a ivestigatio of a equivalece relatio o the itegers that was itroduced as a exercise i our Discrete Math class. Part I - Itro Defiitio Two itegers are Square-Cogruet
More informationSOME TRIBONACCI IDENTITIES
Mathematics Today Vol.7(Dec-011) 1-9 ISSN 0976-38 Abstract: SOME TRIBONACCI IDENTITIES Shah Devbhadra V. Sir P.T.Sarvajaik College of Sciece, Athwalies, Surat 395001. e-mail : drdvshah@yahoo.com The sequece
More informationThe Growth of Functions. Theoretical Supplement
The Growth of Fuctios Theoretical Supplemet The Triagle Iequality The triagle iequality is a algebraic tool that is ofte useful i maipulatig absolute values of fuctios. The triagle iequality says that
More informationTechnical Proofs for Homogeneity Pursuit
Techical Proofs for Homogeeity Pursuit bstract This is the supplemetal material for the article Homogeeity Pursuit, submitted for publicatio i Joural of the merica Statistical ssociatio. B Proofs B. Proof
More informationON THE NUMBER OF LAPLACIAN EIGENVALUES OF TREES SMALLER THAN TWO. Lingling Zhou, Bo Zhou* and Zhibin Du 1. INTRODUCTION
TAIWANESE JOURNAL OF MATHEMATICS Vol 19, No 1, pp 65-75, February 015 DOI: 1011650/tjm190154411 This paper is available olie at http://jouraltaiwamathsocorgtw ON THE NUMBER OF LAPLACIAN EIGENVALUES OF
More informationa for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a
Math E-2b Lecture #8 Notes This week is all about determiats. We ll discuss how to defie them, how to calculate them, lear the allimportat property kow as multiliearity, ad show that a square matrix A
More informationHoggatt and King [lo] defined a complete sequence of natural numbers
REPRESENTATIONS OF N AS A SUM OF DISTINCT ELEMENTS FROM SPECIAL SEQUENCES DAVID A. KLARNER, Uiversity of Alberta, Edmoto, Caada 1. INTRODUCTION Let a, I deote a sequece of atural umbers which satisfies
More informationTHE UNIVERSITY OF TORONTO UNDERGRADUATE MATHEMATICS COMPETITION. In Memory of Robert Barrington Leigh. March 9, 2014
THE UNIVERSITY OF TORONTO UNDERGRADUATE MATHEMATICS COMPETITION I Memory of Robert Barrigto Leigh March 9, 4 Time: 3 hours No aids or calculators permitted. The gradig is desiged to ecourage oly the stroger
More information