Hankel Operators plus Orthogonal Polynomials. Yield Combinatorial Identities
|
|
- Cecilia Tucker
- 6 years ago
- Views:
Transcription
1 Hanel Operators plus Orthogonal Polynomials Yield Combinatorial Identities E. A. Herman, Grinnell College Abstract: A Hanel operator H [h i+j ] can be factored as H MM, where M maps a space of L functions to the corresponding moment sequences. urthermore, a necessary and sufficient condition for a sequence to be in the range of M can be expressed in terms of an expansion in orthogonal polynomials. Combining these two results yields a wealth of combinatorial identities that incorporate both the matrix entries h i+j and the coefficients of the orthogonal polynomials. Mathematics Subject Classification: 47B35, 33D45, 5A19. Keywords: Hanel operators, Hausdorff moments, orthogonal polynomials, combinatorial identities, Hilbert matrices, Legendre polynomials, ibonacci numbers. 1. Introduction The context throughout this paper will be the Hilbert spaces l of square-summable sequences and L α (, 1 of square-integrable functions on [, 1] with the distribution dα, α nondecreasing. Their inner products are the usual dot product for the former and the usual integral inner product for the latter. The Hanel operators in Section are exactly as in [9]. That is, we study the infinite matrices H [h i+j ] i,j, where h j The following result in [9] will be essential: x j dα(x, j, 1,,... Widom s Theorem: The following are equivalent: (a H represents a bounded operator on l. (b h j O(1/j as j. (c α(1 α(x O(1 x as x 1, α(x α( O(1 + x as x +. 1
2 In Theorem 1 we show that H MM, where M : L α (, 1 l is the bounded operator that maps each function f L α(, 1 to its sequence of moments: M(f x f(x dα(x,, 1,,... We also show that, for u u l, we have the formula M u(x u x In Section 3, we explore the space M α of Hausdorff moment sequences given by the range of the operator M. Extending an idea in [], we derive in Theorem the following necessary and sufficient condition for a sequence to belong to M α. If u u j l and p is an orthogonal sequence of polynomials in L α(, 1 with p (x a j x j, a,, 1,,... (1 define the sequence c c by c a j u j,, 1,,... ( Then u u j M α if and only if c / p <. inally, in Theorem 3 we use Theorems 1 and to produce combinatorial identities as follows. Suppose v v l and u Hv. If p and c are the sequences described above in (1 and (, we obtain one type of identity by equating two ways of expressing the c s. On the one hand, c a j (Hv j a j h j+i v i i On the other hand, the c s are ourier coefficients with respect to the orthogonal polynomials p for some function f L α (, 1. Since Hv MM v Mf, where f(x M v(x v x, we obtain a second formula for c in terms of the components v of v. Other identities come from Parseval s equation. In Section 4, we explore several examples.
3 . A factorization of Hanel Operators The Hanel operators we investigate are given by the infinite matrices H [h i+j ] i,j, where h j x j dα(x, j, 1,,... and α is nondecreasing on [, 1]. We assume H represents a bounded operator on l, which means we have the equivalent conditions on h j and α in Widom s theorem. Theorem 1: or all f L α(, 1, let M(f u f u f, where uf x f(x dα(x,, 1,,... (3 or all v v l, let N(v f v, where f v (x v x, 1 < x < 1 (4 (a N is a bounded linear operator from l into L α(, 1. (b M is a bounded linear operator from L α (, 1 into l. (c M N. (d H MM. (e M is injective. Proof : (a We use the Cauchy criterion to show that f v L α (, 1: n ( 1 n ( n n n v x x v j x m mv j dα(x v v j jm m jm n n n n p v v j h j+ v v j + j m jm m jm x +j dα(x for some p >, where the inequality uses the fact that h j O(1/j. Hence, by Hilbert s inequality, v x converges in L α(, 1. 3
4 To show that N is bounded, we use the change of variables g(x dα(x Thus f v g( y dβ(y, where g is integrable and β(y α( y (5 f v (x dα(x + f v ( y dβ(y and the matrices A and B defined below are bounded Hanel operators on l by Widom s Theorem. A [a j+ ], B [b j+ ], where a j Therefore f v (x dα(x x j dα(x, b j y j dβ(y v v j x +j dα(x v v j a +j Av v A v and f v ( y dβ(y v v j ( y +j dβ(y ( v ( j v j b +j B v The fact that v x converges in L α (, 1 justifies the above interchanges of limits. Therefore N(v ( A + B v. (b Again using the change of variables (5, we have, for any f L α(, 1, u f x f(x dα(x x f(x dα(x + ( y f( ydβ(y Let a denote the first of the two integrals on the right and b denote the second. The following argument is an adaptation of the proof in [5, Theorem 34]. Let v v l. The interchange of limits below is again justified by the fact that v x converges in L α (, 1: v a v x f(x dα(x Nv(xf(x dα(x 1 4
5 Hence, by the Cauchy-Schwarz inequality and part (a, v a N(v f N v f Therefore, by the so-called converse of Hölder s inequality [5, Theorem 15], a N f. Similarly, b N f, and so M(f N f. (c and (d are straightforward verifications. (e Suppose M(g. Then, for all v l, M(g v g, M v g, f v Since the functions f v include all polynomials, and since the polynomials are dense in L α (, 1, it follows that g. Although the factorization H MM appears to be new, it is closely related to the wellnown fact [7, Chapter ] that H is unitarily equivalent to the integral transform T on L α (, 1 defined by f(y Tf(x 1 xy dα(y Specifically, one can show that T M M and hence that H and T are unitarily equivalent whenever M is injective. or M to be injective, it suffices to assume that α has infinitely many points of increase with a cluster point in the open interval (, Orthogonal polynomials and moment sequences In this section we use a complete orthogonal sequence p of polynomials in L α (, 1 to characterize the elements of M α, where M α denotes the space of moment sequences given by the range of M (see [8]. Although these sequences may be finite or infinite, to simplify notation, the details below will be expressed as if the sequences are infinite. The conclusions are valid in all cases, and the finite case will be treated as one of the examples in Section 4. The following theorem extends an idea in [], where it was applied to the shifted Legendre polynomials. 5
6 Theorem : Let p be an orthogonal sequence of polynomials in L α (, 1 with p (x a j x j, a,, 1,,... (6 Let u u j l and define c c by c a j u j,, 1,,... (7 Then u M α if and only if urthermore, when (8 holds, then u M(f, where c p < (8 f(x c p p (x (9 Proof : If u u M α, then there exists f L α(, 1 such that u M(f; that is, u x f(x dα(x,, 1,,... Hence, since p is a complete orthogonal sequence in L α (, 1 (see [8], f(x b p (x where b 1 f(xp p (x dα(x Therefore, by equations (7 and (6, c a j x j f(x dα(x f(xp (x dα(x p b So, by Parseval s equation, c p p b f < urthermore, (9 holds since b c / p. 6
7 On the other hand, if u u l and c / p <, let f(x c p p (x Since c / p <, the above series converges in L α (, 1 and c f(xp (x dα(x,, 1,,... This equation can also be written, using (6 and (7, as a j u j which implies, since a, u j f(x a j x j dα(x,, 1,,... f(xx j dα(x, j, 1,,... Therefore u u j M(f M α. inally, we use Theorems 1 and to produce combinatorial identities corresponding to each choice of α. Theorem 3: Suppose α is a nondecreasing function on [, 1] satisfying and let α(1 α(x O(1 x as x 1, α(x α( O(1 + x as x + h j x j dα(x, j, 1,,... Suppose also that p is an orthogonal sequence of polynomials in L α (, 1 with p (x a j x j, a,, 1,,... and x b j p j (x,, 1,,... 7
8 Then, for all non-negative integers and m, { p b m if m a j h j+m if m < (1 and m i urthermore, for all v v l, 1 p i 1 p i i a ij h j+m a i δ m (11 a j h j+i v i v v j h +j (1 and p v j b j j v v j h +j (13 Proof : Let H [h i+j ] i,j, v v l, and u Hv MM v. Then u M α, and we use Theorem to find two expressions for the coefficients c in equation (7. On the one hand, c a j u j a j (Hv j a j h j+i v i (14 On the other hand, i c f(xp (x dα(x where f(x c p p (x Using Theorem 1, we may also write the last expression for c in terms of the components of v. Note that Hv M(f and Hv MM v M(f v. Since M is injective, we have f f v a. e. and so c v j x j p (x dα(x v j i j b ji p i (xp (x dα(x p v j b j (15 j 8
9 Equating formulas (14 and (15 for c and letting v δ m, we obtain the identity (1. Then, solving for b m, (m in (1 and substituting into m i b ma i δ mi (m i, we obtain the identity (11. urthermore, by Parseval s equation and MM H, we have c p f v M v M v, M v Hv v v v j h +j Substituting formulas (14 and (15 for c in this result yields the identities (1 and (13. Another perspective on Theorem 3 The analysis below shows that, with a different set of assumptions, the identities in Theorem 3 are quite natural. Thus, some instances of them are liely to be nown. Suppose, instead of a distribution dα as in Theorem 3, we are given the following assumptions about a matrix H and a sequence of polynomials p. Let H [h i+j ] i,j be any infinite Hanel matrix with real entries; that is, do not assume h j O(1/j as j. Also let p (x a j x j, a,, 1,,... be a sequence of polynomials with real coefficients that are orthogonal in the sense that { m if m a i h i+j a mj (16 if m i Note that, under the assumptions of Theorem 3, property (16 is the statement that p, p m if m and p, p. So we will denote the left side of (16 by p, p m even though the matrix H does not necessarily generate a true inner product. Then (16 can be written as AHA D, where A denotes the lower-triangular matrix whose (, j entry, for j, is a j and D denotes the diagonal matrix whose th diagonal entry is p, p. Since the diagonal entries of A are nonzero, A has a formal inverse B. Hence AHA D implies AH DB, which is the identity (1. Of course, p must be interpreted as p, p, which is nonzero but possibly negative. Identity (11 follows from (1, as in the proof of Theorem 3. The left side of identity (1 can be expressed as D / AHv, where D / is diagonal and the norm is the usual l norm. However, now we must assume that the sequence v has only finitely many nonzero terms. Then (1 may be derived as follows: D / AHv (D / AHv (D / AHv (D DB v (AHv v Hv 9
10 Identity (13 may be derived similarly. 4. Examples In the examples below, some of the instances of identity (1 have been independently confirmed using the WZ methods in [6]. On the other hand, instances of identities (1 and (13 do not yet seem to be susceptible of confirmation by such algorithmic methods. Example 1 (Hilbert matrices and binomial coefficients: or any r >, let α(x x r if x 1 and α(x if x. Then h j r/(j +r and so H [r/(i+j+r] i,j. In [3], Berg shows that the corresponding orthogonal polynomials may be written as p (x ( ( + j + r 1 ( j x j j j It follows that p r/( + r and x 1 ( + r ( +r ( j ( + r j (j + rp j (x Hence equations (1, (11, (1 and (13 become, respectively, ( ( ( + j + r 1 ( j m+r j j + m + r ( m ( m+r (m + r m i i ( ( ( ( i i + j + r 1 i i + + r 1 ( j+ (i + 1 δ m j i i j + m + r where v j l. ( ( + j + r 1 ( j v i v v j ( + r j j + i + r + j + r i ( j+r j vj ( + r (j + r ( v v j j+r + j + r j j m Example (Legendre polynomials: We begin by considering all Jacobi polynomials: p (α,β (x (α (, α + β + + 1; α + 1; (1 x/! 1
11 where 1 denotes the Gauss hypergeometric series, (α + 1 denotes a shifted factorial, and α, β >. The Jacobi polynomials are orthogonal on [, 1] relative to the weight function w(x (1 x α (1 + x β, and p (α,β α+β+1 Γ(α + + 1Γ(β + + 1! Γ(α + β + + 1(α + β Using Euler s integral representation of Gauss hypergeometric series [1, Theorem..1], we find the moments to be h j x j (1 x α (1 + x β dx ( j! ( j 1 ( α, 1 + j; + β + j; + 1 ( β, 1 + j; + α + j; (1 + β 1+j (1 + α 1+j Rather than writing out identities (1, (11, (1, and (13 for this general case, we write the more compact versions for the special case α β. Then h j /(j +1 if j is even, h j if j is odd, and p p (, is the th Legendre polynomial; hence p /(+1 and ( ( ( ( j/ + j p (x j x j x j+ even j+ even!(j + 1 ( j/ ( 1 ( j!(j + + 1!!p j(x Note: The double factorial n!! is defined to be 1 when n and, for positive integers n, n!! n(n (n 4 (1 or where the terminal factor in this product is 1 when n is odd and when n is even. Hence equations (1, (11, (1, and (13 become, respectively, ( ( { + j ( ( j/ m! j (j + m + 1 if m + even, m (m / ( 1 (m!(+m+1!! otherwise j+ even j+m even m i i+ even i i+j even j+m even ( ( i (j+/ i i i j (j + m ( ( ( i + j i i + i δ m i i
12 ( ( ( j ( j/ v i j i (j + i + 1 j+ even j+i even ( + 1 j!v j (j / ( 1 (j!( + j + 1!! j where v j l. j+ even j+ even j+ even v i v j + j + 1 v i v j + j + 1 Example 3 (ibonacci numbers and ibonomial coefficients: We begin by considering all little q-jacobi polynomials [4, Section 7.3]: p (x; a, b; q n (q ; q j (abq +1 ; q j (xq j (q; q j (aq; q j where (a; q j denotes the q-shifted factorial. We assume that q < 1, aq < 1, and b 1, which suffices to show that these polynomials satisfy the orthogonality property where i p (q i ; a, b; qp m (q i ; a, b; q (bq; q i (q; q i (aq i δ m h m (a, b; q h m (a, b; q (abq; q m(1 abq m+1 (aq; q m (aq; q (aq m (q; q m (1 abq(bq; q m (abq ; q We can express the orthogonality property (17 in terms of an integral on [, 1] as follows. Let p (x p (φx; a, b; q, where φ > 1, and define the discrete signed measure µ by (17 µ i (bq; q i (q; q i (aq i δ q i /φ (18 Then (17 becomes The corresponding moments are h j p (xp m (x dµ(x x j dµ(x i 1 δ m h m (a, b; q (bq; q i (q; q i (aq i ( q i /φ j
13 When b 1, this sum is a geometric series: h j 1 φ j i ( aq j+1 i 1 φ j (1 aq j+1 The following interesting special case was found by Berg [3]. or any α N, let b 1, φ 1 + 5, q , a qα, and µ α (1 q α q iα δ q i /φ, α 1,, 3,... i Note that µ α is the signed measure (18 multiplied by 1 q α so it becomes a measure with total mass 1. Berg shows that h j α / α+j, j, 1,,..., that p (α (x p (φx; q α, 1; q ( ( ( j (j α + + j 1 j x j and p (α (xp m (α (x dµ α (x δ m ( α α α+ where j denotes the jth ibonacci number (with, 1 1 and ( ibonomial coefficient j i+1 i1 i. j denotes the When α is even, µ α is a probability measure that, together with the polynomials p (α (x, satisfies the hypotheses of Theorem 3. Note that p (α α / α+. Hence, equations (11 and (1 become m i i ( ( i(j+ (j ( i j α+ i ( ( i α + i + j 1 i ( ( ( j (j α + + j 1 j ( α + i + 1 vi α+j+i i α+i α+j+m δ m v v j α++j where α, 4, 6,... and v j l. urthermore, the analysis following Theorem 3 shows that both identities also hold for α 1, 3, 5,..., provided v j for all but finitely many values of j. or such α, note that p (α, p (α < when is odd. 13
14 Example 4 (finite dimensional spaces: Let α be a step function with positive jumps J at the distinct points z (, 1,, 1,..., r. Then L α(, 1 has dimension r + 1, and r h j z j J, j, 1,,... (19 In what follows, we will frequently use the following submatrices of the infinite Hanel matrix H [h i+j ] i,j : Let H denote the upper-left ( + 1 ( + 1 submatrix of H. rom equation (19, we see that H V D r V T,, 1,..., r ( where V is rows through of the Vandermonde matrix whose row 1 is [z, z 1,...,z r ], and where D r is the diagonal matrix with diagonal entries J, J 1,...,J r. In seeing an orthogonal basis of polynomials for L α(, 1, we choose their leading coefficients to be 1: p (x x + a j x j,, 1,..., r The remaining coefficients are then uniquely determined; we compute them as follows. Let h(p, q denote the vector with components h p,...,h q, and let a( denote the vector with components a,...,a,. We claim that which may also be written H a( h(, 1, 1,...,r (1 h m+j a j h m+, 1,...,r, m,..., 1 Since H is nonsingular when, 1,...,r, equation (1 uniquely determines the coefficients a j. One may confirm (1 by a straightforward calculation to show that the resulting polynomials are mutually orthogonal. By Cramer s rule, we then have the explicit formula p (x x 1 det(h (j,h(, 1x j,, 1,..., r ( deth where M(j,w denotes the matrix M with its jth column replaced by the vector w. A similar calculation shows that p deth deth,, 1,..., r 14
15 where we assign the value 1 to deth. Next we express each monomial x m as a linear combination of the basis polynomials p, p 1,...,p r : r x m b m p (x, m, 1,,... By equations (1 and (, b m 1 p a j h j+m ( deth 1 h +m det(h (j,h(, 1h j+m deth deth ( 1 (deth h +m det(h (j,h(, 1h j+m deth det(h (,h(m, m + deth To confirm the last step above, evaluate det(h (,h(m, m+ by the Laplace expansion down the last column of the matrix. or the jth entry of that column, j, 1,..., 1, the expansion formula assigns the sign ( +j. We must also move the vector h(, from column j to column 1, and the corresponding transposes change the sign of the determinant by the factor ( j. Thus, the overall sign factor is ( +j+ j. Having confirmed the above computation of b m, we conclude x m r det(h (,h(m, m + deth p (x, m, 1,,... Hence equation (13 becomes r 1 (deth (deth v j det(h (,h(j, j + j v v j h +j where v j l. Acnowledgment: I than my colleague C. rench for suggesting the analysis following Theorem 3. 15
16 References 1. G. E. Andrews, R. Asey, R. Roy, Special unctions, Cambridge University Press, R. Asey, I. J. Schoenberg, A. Sharma, Hausdorff s moment problem and expansions in Legendre polynomials, J. Math. Anal. Appl., v. 86 (198, C. Berg, ibonacci numbers and orthogonal polynomials, J. Comput. Appl. Math. (to appear. 4. G. Gasper and M. Rahman, Basic Hypergeometric Series, nd ed., Cambridge University Press, G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, M. Petovše, H. Wilf, D. Zeilberger, A B, A. K. Peters, Ltd., S. C. Power, Hanel Operators on Hilbert Space, Pitman Publishing, Inc., G. Szegö, Orthogonal Polynomials, 4th ed., Amer. Math. Soc., H. Widom, Hanel Matrices, Trans. Amer. Math. Soc., v. 11 (1966,
DS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More informationCombinatorial Analysis of the Geometric Series
Combinatorial Analysis of the Geometric Series David P. Little April 7, 205 www.math.psu.edu/dlittle Analytic Convergence of a Series The series converges analytically if and only if the sequence of partial
More informationLinear Algebra V = T = ( 4 3 ).
Linear Algebra Vectors A column vector is a list of numbers stored vertically The dimension of a column vector is the number of values in the vector W is a -dimensional column vector and V is a 5-dimensional
More informationStrictly Positive Definite Functions on a Real Inner Product Space
Strictly Positive Definite Functions on a Real Inner Product Space Allan Pinkus Abstract. If ft) = a kt k converges for all t IR with all coefficients a k 0, then the function f< x, y >) is positive definite
More informationMAT Linear Algebra Collection of sample exams
MAT 342 - Linear Algebra Collection of sample exams A-x. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system
More informationANALOGUES OF THE TRIPLE PRODUCT IDENTITY, LEBESGUE S IDENTITY AND EULER S PENTAGONAL NUMBER THEOREM
q-hypergeometric PROOFS OF POLYNOMIAL ANALOGUES OF THE TRIPLE PRODUCT IDENTITY, LEBESGUE S IDENTITY AND EULER S PENTAGONAL NUMBER THEOREM S OLE WARNAAR Abstract We present alternative, q-hypergeometric
More informationMatrices. Chapter What is a Matrix? We review the basic matrix operations. An array of numbers a a 1n A = a m1...
Chapter Matrices We review the basic matrix operations What is a Matrix? An array of numbers a a n A = a m a mn with m rows and n columns is a m n matrix Element a ij in located in position (i, j The elements
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 informationClass notes: Approximation
Class notes: Approximation Introduction Vector spaces, linear independence, subspace The goal of Numerical Analysis is to compute approximations We want to approximate eg numbers in R or C vectors in R
More informationACI-matrices all of whose completions have the same rank
ACI-matrices all of whose completions have the same rank Zejun Huang, Xingzhi Zhan Department of Mathematics East China Normal University Shanghai 200241, China Abstract We characterize the ACI-matrices
More informationVectors in Function Spaces
Jim Lambers MAT 66 Spring Semester 15-16 Lecture 18 Notes These notes correspond to Section 6.3 in the text. Vectors in Function Spaces We begin with some necessary terminology. A vector space V, also
More informationLinear Algebra. Session 12
Linear Algebra. Session 12 Dr. Marco A Roque Sol 08/01/2017 Example 12.1 Find the constant function that is the least squares fit to the following data x 0 1 2 3 f(x) 1 0 1 2 Solution c = 1 c = 0 f (x)
More informationFunctional Analysis Review
Outline 9.520: Statistical Learning Theory and Applications February 8, 2010 Outline 1 2 3 4 Vector Space Outline A vector space is a set V with binary operations +: V V V and : R V V such that for all
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationMath Linear Algebra II. 1. Inner Products and Norms
Math 342 - Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,
More informationLinear Algebra March 16, 2019
Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented
More informationTwo finite forms of Watson s quintuple product identity and matrix inversion
Two finite forms of Watson s uintuple product identity and matrix inversion X. Ma Department of Mathematics SuZhou University, SuZhou 215006, P.R.China Submitted: Jan 24, 2006; Accepted: May 27, 2006;
More informationLecture 1: Review of linear algebra
Lecture 1: Review of linear algebra Linear functions and linearization Inverse matrix, least-squares and least-norm solutions Subspaces, basis, and dimension Change of basis and similarity transformations
More informationMathematics Department Stanford University Math 61CM/DM Inner products
Mathematics Department Stanford University Math 61CM/DM Inner products Recall the definition of an inner product space; see Appendix A.8 of the textbook. Definition 1 An inner product space V is a vector
More informationVector Spaces. Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms.
Vector Spaces Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms. For each two vectors a, b ν there exists a summation procedure: a +
More informationRemarks on the Rademacher-Menshov Theorem
Remarks on the Rademacher-Menshov Theorem Christopher Meaney Abstract We describe Salem s proof of the Rademacher-Menshov Theorem, which shows that one constant works for all orthogonal expansions in all
More informationOR MSc Maths Revision Course
OR MSc Maths Revision Course Tom Byrne School of Mathematics University of Edinburgh t.m.byrne@sms.ed.ac.uk 15 September 2017 General Information Today JCMB Lecture Theatre A, 09:30-12:30 Mathematics revision
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationReview of Basic Concepts in Linear Algebra
Review of Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University September 7, 2017 Math 565 Linear Algebra Review September 7, 2017 1 / 40 Numerical Linear Algebra
More informationStatistics 612: L p spaces, metrics on spaces of probabilites, and connections to estimation
Statistics 62: L p spaces, metrics on spaces of probabilites, and connections to estimation Moulinath Banerjee December 6, 2006 L p spaces and Hilbert spaces We first formally define L p spaces. Consider
More informationDeterminant formulas for multidimensional hypergeometric period matrices
Advances in Applied Mathematics 29 (2002 137 151 www.academicpress.com Determinant formulas for multidimensional hypergeometric period matrices Donald Richards a,b,,1 andqifuzheng c,2 a School of Mathematics,
More informationSome generalizations of a supercongruence of van Hamme
Some generalizations of a supercongruence of van Hamme Victor J. W. Guo School of Mathematical Sciences, Huaiyin Normal University, Huai an, Jiangsu 3300, People s Republic of China jwguo@hytc.edu.cn Abstract.
More informationj=1 [We will show that the triangle inequality holds for each p-norm in Chapter 3 Section 6.] The 1-norm is A F = tr(a H A).
Math 344 Lecture #19 3.5 Normed Linear Spaces Definition 3.5.1. A seminorm on a vector space V over F is a map : V R that for all x, y V and for all α F satisfies (i) x 0 (positivity), (ii) αx = α x (scale
More informationDeterminants Chapter 3 of Lay
Determinants Chapter of Lay Dr. Doreen De Leon Math 152, Fall 201 1 Introduction to Determinants Section.1 of Lay Given a square matrix A = [a ij, the determinant of A is denoted by det A or a 11 a 1j
More informationHW2 Solutions
8.024 HW2 Solutions February 4, 200 Apostol.3 4,7,8 4. Show that for all x and y in real Euclidean space: (x, y) 0 x + y 2 x 2 + y 2 Proof. By definition of the norm and the linearity of the inner product,
More informationMathematical Foundations
Chapter 1 Mathematical Foundations 1.1 Big-O Notations In the description of algorithmic complexity, we often have to use the order notations, often in terms of big O and small o. Loosely speaking, for
More informationFrame Diagonalization of Matrices
Frame Diagonalization of Matrices Fumiko Futamura Mathematics and Computer Science Department Southwestern University 00 E University Ave Georgetown, Texas 78626 U.S.A. Phone: + (52) 863-98 Fax: + (52)
More informationChapter 1. Preliminaries. The purpose of this chapter is to provide some basic background information. Linear Space. Hilbert Space.
Chapter 1 Preliminaries The purpose of this chapter is to provide some basic background information. Linear Space Hilbert Space Basic Principles 1 2 Preliminaries Linear Space The notion of linear space
More information(K + L)(c x) = K(c x) + L(c x) (def of K + L) = K( x) + K( y) + L( x) + L( y) (K, L are linear) = (K L)( x) + (K L)( y).
Exercise 71 We have L( x) = x 1 L( v 1 ) + x 2 L( v 2 ) + + x n L( v n ) n = x i (a 1i w 1 + a 2i w 2 + + a mi w m ) i=1 ( n ) ( n ) ( n ) = x i a 1i w 1 + x i a 2i w 2 + + x i a mi w m i=1 Therefore y
More informationStrictly positive definite functions on a real inner product space
Advances in Computational Mathematics 20: 263 271, 2004. 2004 Kluwer Academic Publishers. Printed in the Netherlands. Strictly positive definite functions on a real inner product space Allan Pinkus Department
More informationReview of Linear Algebra
Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=
More informationLinear Algebra and Matrix Inversion
Jim Lambers MAT 46/56 Spring Semester 29- Lecture 2 Notes These notes correspond to Section 63 in the text Linear Algebra and Matrix Inversion Vector Spaces and Linear Transformations Matrices are much
More informationThen x 1,..., x n is a basis as desired. Indeed, it suffices to verify that it spans V, since n = dim(v ). We may write any v V as r
Practice final solutions. I did not include definitions which you can find in Axler or in the course notes. These solutions are on the terse side, but would be acceptable in the final. However, if you
More informationMATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product.
MATH 311-504 Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product. Determinant is a scalar assigned to each square matrix. Notation. The determinant of a matrix A = (a ij
More informationTransformations Preserving the Hankel Transform
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol 10 (2007), Article 0773 Transformations Preserving the Hankel Transform Christopher French Department of Mathematics and Statistics Grinnell College Grinnell,
More informationChapter SSM: Linear Algebra Section Fails to be invertible; since det = 6 6 = Invertible; since det = = 2.
SSM: Linear Algebra Section 61 61 Chapter 6 1 2 1 Fails to be invertible; since det = 6 6 = 0 3 6 3 5 3 Invertible; since det = 33 35 = 2 7 11 5 Invertible; since det 2 5 7 0 11 7 = 2 11 5 + 0 + 0 0 0
More informationUsing q-calculus to study LDL T factorization of a certain Vandermonde matrix
Using q-calculus to study LDL T factorization of a certain Vandermonde matrix Alexey Kuznetsov May 2, 207 Abstract We use tools from q-calculus to study LDL T decomposition of the Vandermonde matrix V
More informationON THE HÖLDER CONTINUITY OF MATRIX FUNCTIONS FOR NORMAL MATRICES
Volume 10 (2009), Issue 4, Article 91, 5 pp. ON THE HÖLDER CONTINUITY O MATRIX UNCTIONS OR NORMAL MATRICES THOMAS P. WIHLER MATHEMATICS INSTITUTE UNIVERSITY O BERN SIDLERSTRASSE 5, CH-3012 BERN SWITZERLAND.
More informationKernel Method: Data Analysis with Positive Definite Kernels
Kernel Method: Data Analysis with Positive Definite Kernels 2. Positive Definite Kernel and Reproducing Kernel Hilbert Space Kenji Fukumizu The Institute of Statistical Mathematics. Graduate University
More informationA property concerning the Hadamard powers of inverse M-matrices
Linear Algebra and its Applications 381 (2004 53 60 www.elsevier.com/locate/laa A property concerning the Hadamard powers of inverse M-matrices Shencan Chen Department of Mathematics, Fuzhou University,
More informationMATH2210 Notebook 2 Spring 2018
MATH2210 Notebook 2 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 2 MATH2210 Notebook 2 3 2.1 Matrices and Their Operations................................
More informationBasic Concepts in Linear Algebra
Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University February 2, 2015 Grady B Wright Linear Algebra Basics February 2, 2015 1 / 39 Numerical Linear Algebra Linear
More informationHilbert Spaces. Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space.
Hilbert Spaces Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space. Vector Space. Vector space, ν, over the field of complex numbers,
More informationMath Linear Algebra Final Exam Review Sheet
Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of
More informationMatrix Algebra: Vectors
A Matrix Algebra: Vectors A Appendix A: MATRIX ALGEBRA: VECTORS A 2 A MOTIVATION Matrix notation was invented primarily to express linear algebra relations in compact form Compactness enhances visualization
More informationEE731 Lecture Notes: Matrix Computations for Signal Processing
EE731 Lecture Notes: Matrix Computations for Signal Processing James P. Reilly c Department of Electrical and Computer Engineering McMaster University September 22, 2005 0 Preface This collection of ten
More informationMath 321: Linear Algebra
Math 32: Linear Algebra T. Kapitula Department of Mathematics and Statistics University of New Mexico September 8, 24 Textbook: Linear Algebra,by J. Hefferon E-mail: kapitula@math.unm.edu Prof. Kapitula,
More informationChapter 1 Vector Spaces
Chapter 1 Vector Spaces Per-Olof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 110 Linear Algebra Vector Spaces Definition A vector space V over a field
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationLemma 8: Suppose the N by N matrix A has the following block upper triangular form:
17 4 Determinants and the Inverse of a Square Matrix In this section, we are going to use our knowledge of determinants and their properties to derive an explicit formula for the inverse of a square matrix
More informationChapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1
More information1 Determinants. 1.1 Determinant
1 Determinants [SB], Chapter 9, p.188-196. [SB], Chapter 26, p.719-739. Bellow w ll study the central question: which additional conditions must satisfy a quadratic matrix A to be invertible, that is to
More informationChapter 4 No. 4.0 Answer True or False to the following. Give reasons for your answers.
MATH 434/534 Theoretical Assignment 3 Solution Chapter 4 No 40 Answer True or False to the following Give reasons for your answers If a backward stable algorithm is applied to a computational problem,
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationInner Product Spaces An inner product on a complex linear space X is a function x y from X X C such that. (1) (2) (3) x x > 0 for x 0.
Inner Product Spaces An inner product on a complex linear space X is a function x y from X X C such that (1) () () (4) x 1 + x y = x 1 y + x y y x = x y x αy = α x y x x > 0 for x 0 Consequently, (5) (6)
More informationLecture 7. Econ August 18
Lecture 7 Econ 2001 2015 August 18 Lecture 7 Outline First, the theorem of the maximum, an amazing result about continuity in optimization problems. Then, we start linear algebra, mostly looking at familiar
More informationFundamentals of Linear Algebra. Marcel B. Finan Arkansas Tech University c All Rights Reserved
Fundamentals of Linear Algebra Marcel B. Finan Arkansas Tech University c All Rights Reserved October 9, 200 2 PREFACE Linear algebra has evolved as a branch of mathematics with wide range of applications
More informationNOTES on LINEAR ALGEBRA 1
School of Economics, Management and Statistics University of Bologna Academic Year 207/8 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura
More informationEECS 598: Statistical Learning Theory, Winter 2014 Topic 11. Kernels
EECS 598: Statistical Learning Theory, Winter 2014 Topic 11 Kernels Lecturer: Clayton Scott Scribe: Jun Guo, Soumik Chatterjee Disclaimer: These notes have not been subjected to the usual scrutiny reserved
More informationFunctional Analysis Review
Functional Analysis Review Lorenzo Rosasco slides courtesy of Andre Wibisono 9.520: Statistical Learning Theory and Applications September 9, 2013 1 2 3 4 Vector Space A vector space is a set V with binary
More informationMath General Topology Fall 2012 Homework 1 Solutions
Math 535 - General Topology Fall 2012 Homework 1 Solutions Definition. Let V be a (real or complex) vector space. A norm on V is a function : V R satisfying: 1. Positivity: x 0 for all x V and moreover
More informationTOPOLOGICAL COMPLEXITY OF 2-TORSION LENS SPACES AND ku-(co)homology
TOPOLOGICAL COMPLEXITY OF 2-TORSION LENS SPACES AND ku-(co)homology DONALD M. DAVIS Abstract. We use ku-cohomology to determine lower bounds for the topological complexity of mod-2 e lens spaces. In the
More informationAsymptotics of Integrals of. Hermite Polynomials
Applied Mathematical Sciences, Vol. 4, 010, no. 61, 04-056 Asymptotics of Integrals of Hermite Polynomials R. B. Paris Division of Complex Systems University of Abertay Dundee Dundee DD1 1HG, UK R.Paris@abertay.ac.uk
More informationWeek 9-10: Recurrence Relations and Generating Functions
Week 9-10: Recurrence Relations and Generating Functions April 3, 2017 1 Some number sequences An infinite sequence (or just a sequence for short is an ordered array a 0, a 1, a 2,..., a n,... of countably
More informationAlgebra II. Paulius Drungilas and Jonas Jankauskas
Algebra II Paulius Drungilas and Jonas Jankauskas Contents 1. Quadratic forms 3 What is quadratic form? 3 Change of variables. 3 Equivalence of quadratic forms. 4 Canonical form. 4 Normal form. 7 Positive
More informationOn a restriction problem of de Leeuw type for Laguerre multipliers
On a restriction problem of de Leeuw type for Laguerre multipliers George Gasper 1 and Walter Trebels 2 Dedicated to Károly Tandori on the occasion of his 7th birthday (Published in Acta Math. Hungar.
More informationSome results on the existence of t-all-or-nothing transforms over arbitrary alphabets
Some results on the existence of t-all-or-nothing transforms over arbitrary alphabets Navid Nasr Esfahani, Ian Goldberg and Douglas R. Stinson David R. Cheriton School of Computer Science University of
More informationBP -HOMOLOGY AND AN IMPLICATION FOR SYMMETRIC POLYNOMIALS. 1. Introduction and results
BP -HOMOLOGY AND AN IMPLICATION FOR SYMMETRIC POLYNOMIALS DONALD M. DAVIS Abstract. We determine the BP -module structure, mod higher filtration, of the main part of the BP -homology of elementary 2- groups.
More informationNATIONAL BOARD FOR HIGHER MATHEMATICS. Research Scholarships Screening Test. Saturday, February 2, Time Allowed: Two Hours Maximum Marks: 40
NATIONAL BOARD FOR HIGHER MATHEMATICS Research Scholarships Screening Test Saturday, February 2, 2008 Time Allowed: Two Hours Maximum Marks: 40 Please read, carefully, the instructions on the following
More informationMATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
More informationComplete Nevanlinna-Pick Kernels
Complete Nevanlinna-Pick Kernels Jim Agler John E. McCarthy University of California at San Diego, La Jolla California 92093 Washington University, St. Louis, Missouri 63130 Abstract We give a new treatment
More informationMath 3108: Linear Algebra
Math 3108: Linear Algebra Instructor: Jason Murphy Department of Mathematics and Statistics Missouri University of Science and Technology 1 / 323 Contents. Chapter 1. Slides 3 70 Chapter 2. Slides 71 118
More informationMath 5520 Homework 2 Solutions
Math 552 Homework 2 Solutions March, 26. Consider the function fx) = 2x ) 3 if x, 3x ) 2 if < x 2. Determine for which k there holds f H k, 2). Find D α f for α k. Solution. We show that k = 2. The formulas
More informationb jσ(j), Keywords: Decomposable numerical range, principal character AMS Subject Classification: 15A60
On the Hu-Hurley-Tam Conjecture Concerning The Generalized Numerical Range Che-Man Cheng Faculty of Science and Technology, University of Macau, Macau. E-mail: fstcmc@umac.mo and Chi-Kwong Li Department
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 informationAn Involution for the Gauss Identity
An Involution for the Gauss Identity William Y. C. Chen Center for Combinatorics Nankai University, Tianjin 300071, P. R. China Email: chenstation@yahoo.com Qing-Hu Hou Center for Combinatorics Nankai
More informationPositivity of Turán determinants for orthogonal polynomials
Positivity of Turán determinants for orthogonal polynomials Ryszard Szwarc Abstract The orthogonal polynomials p n satisfy Turán s inequality if p 2 n (x) p n 1 (x)p n+1 (x) 0 for n 1 and for all x in
More informationON SUMS OF PRIMES FROM BEATTY SEQUENCES. Angel V. Kumchev 1 Department of Mathematics, Towson University, Towson, MD , U.S.A.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A08 ON SUMS OF PRIMES FROM BEATTY SEQUENCES Angel V. Kumchev 1 Department of Mathematics, Towson University, Towson, MD 21252-0001,
More informationLinear Systems and Matrices
Department of Mathematics The Chinese University of Hong Kong 1 System of m linear equations in n unknowns (linear system) a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.......
More informationMatrix Algebra. Matrix Algebra. Chapter 8 - S&B
Chapter 8 - S&B Algebraic operations Matrix: The size of a matrix is indicated by the number of its rows and the number of its columns. A matrix with k rows and n columns is called a k n matrix. The number
More informationChapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations
Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2
More informationMath 321 Final Examination April 1995 Notation used in this exam: N. (1) S N (f,x) = f(t)e int dt e inx.
Math 321 Final Examination April 1995 Notation used in this exam: N 1 π (1) S N (f,x) = f(t)e int dt e inx. 2π n= N π (2) C(X, R) is the space of bounded real-valued functions on the metric space X, equipped
More informationRecall that any inner product space V has an associated norm defined by
Hilbert Spaces Recall that any inner product space V has an associated norm defined by v = v v. Thus an inner product space can be viewed as a special kind of normed vector space. In particular every inner
More information(1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define
Homework, Real Analysis I, Fall, 2010. (1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define ρ(f, g) = 1 0 f(x) g(x) dx. Show that
More informationANALYSIS QUALIFYING EXAM FALL 2017: SOLUTIONS. 1 cos(nx) lim. n 2 x 2. g n (x) = 1 cos(nx) n 2 x 2. x 2.
ANALYSIS QUALIFYING EXAM FALL 27: SOLUTIONS Problem. Determine, with justification, the it cos(nx) n 2 x 2 dx. Solution. For an integer n >, define g n : (, ) R by Also define g : (, ) R by g(x) = g n
More informationFormula for the inverse matrix. Cramer s rule. Review: 3 3 determinants can be computed expanding by any row or column
Math 20F Linear Algebra Lecture 18 1 Determinants, n n Review: The 3 3 case Slide 1 Determinants n n (Expansions by rows and columns Relation with Gauss elimination matrices: Properties) Formula for the
More informationReview of linear algebra
Review of linear algebra 1 Vectors and matrices We will just touch very briefly on certain aspects of linear algebra, most of which should be familiar. Recall that we deal with vectors, i.e. elements of
More informationMathematics. EC / EE / IN / ME / CE. for
Mathematics for EC / EE / IN / ME / CE By www.thegateacademy.com Syllabus Syllabus for Mathematics Linear Algebra: Matrix Algebra, Systems of Linear Equations, Eigenvalues and Eigenvectors. Probability
More informationMathematical Methods wk 1: Vectors
Mathematical Methods wk : Vectors John Magorrian, magog@thphysoxacuk These are work-in-progress notes for the second-year course on mathematical methods The most up-to-date version is available from http://www-thphysphysicsoxacuk/people/johnmagorrian/mm
More informationMathematical Methods wk 1: Vectors
Mathematical Methods wk : Vectors John Magorrian, magog@thphysoxacuk These are work-in-progress notes for the second-year course on mathematical methods The most up-to-date version is available from http://www-thphysphysicsoxacuk/people/johnmagorrian/mm
More informationA combinatorial determinant
A combinatorial erminant Herbert S Wilf Department of Mathematics, University of Pennsylvania Philadelphia, PA 19104-6395 Abstract A theorem of Mina evaluates the erminant of a matrix with entries D (f(x)
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 informationChapter 3. Linear and Nonlinear Systems
59 An expert is someone who knows some of the worst mistakes that can be made in his subject, and how to avoid them Werner Heisenberg (1901-1976) Chapter 3 Linear and Nonlinear Systems In this chapter
More informationLinear Algebra (Review) Volker Tresp 2017
Linear Algebra (Review) Volker Tresp 2017 1 Vectors k is a scalar (a number) c is a column vector. Thus in two dimensions, c = ( c1 c 2 ) (Advanced: More precisely, a vector is defined in a vector space.
More information